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E„Pf."D„:L$(/L$(X L$(L9,$XT$HD$Y"XD$% -!f(f(fTf.v7H,ffUH*f(f(=* fT\f(fV]T$HH$\!YD$f^$XD$f/wf/D$s;H,fHCH*f.E„f.!D„f(LfH)H*f.cE„f.E!D„T$ T$ f(MfXʺI)ID$H*f.E„'f. D„L$ VL$ J3XfH*f.E„f. D„L$ L$ X4$T$( \\Y\f/s;f($\Yf/3f($ $Xf/HD$8H9D$@ILH)L9l$0HOHXH[]A\A]A^A_DfZfff(XT$ Dfffdff.H ~ ; f.f(f(H8\\T$L$D$(f(\$^l$ 4$4$\$L$T$f/r3l$ fd$(YYf.QwHXH8f(@f(\ x\fYYf.Qw*\H8f(d$$d$$f(T$ $T$ $SHH -D$\f( D$H f/D$$H YD$  $\f(Yf/rXf(T$ T$$f(u  $^dXH,HjH [f/ʸrHHlongan integer is requirednumpy/random/mtrand/mtrand.c%s (%s:%d)Missing type objectnumpy%s.%s is not a type object%s() keywords must be stringsname '%s' is not definedcannot import name %.230smtrand.pyxmtrand.RandomState.__reduce__mtrand.RandomState.randnmtrand.RandomState.randat leastat mostmtrand.RandomState.__init__exactlymtrand._shape_from_sizerandom_integersmtrand.disc0_arraymtrand.RandomState.tomaxintmtrand.discd_array_scmtrand.cont0_arraystandard_cauchystandard_exponentialstandard_normalrandom_samplemtrand.cont1_array_scmtrand.cont2_array_scmtrand.cont3_array_scrandint_helpers.pximtrand._rand_int8mtrand._rand_boolmtrand._rand_uint8mtrand._rand_uint32mtrand._rand_uint16mtrand._rand_uint64mtrand.RandomState.get_statefloat divisionmtrand.RandomState.dirichletmtrand.discnp_arraymtrand.discdd_arraymtrand.discnmN_arraymtrand.discd_arraymtrand.RandomState.zipfmtrand.cont1_arraymtrand.RandomState.rayleighmtrand.RandomState.powermtrand.RandomState.weibullmtrand.RandomState.paretomtrand.RandomState.standard_tmtrand.RandomState.chisquarestandard_gammamtrand.RandomState.set_statemtrand.cont2_arraymtrand.RandomState.lognormalmtrand.RandomState.logisticmtrand.RandomState.gumbelmtrand.RandomState.laplacemtrand.RandomState.vonmisesmtrand.RandomState.normalmtrand.RandomState.uniformmtrand.cont3_arraymtrand.RandomState.geometricmtrand.RandomState.logseriesmtrand.RandomState.poissonmultinomialmtrand.RandomState.betanoncentral_chisquaremtrand.RandomState.fmtrand.RandomState.waldmtrand.RandomState.gammamtrand.RandomState.bytesmtrand.RandomState.triangularhypergeometricmtrand.discnmN_array_scnoncentral_fmtrand.RandomState.shufflemultivariate_normalnegative_binomialmtrand.discdd_array_scmtrand.RandomState.randintmtrand.RandomState.binomialmtrand.discnp_array_scmtrand._rand_int64mtrand._rand_int16mtrand._rand_int32mtrand.RandomState.seedmtrand.RandomState.choiceassignmentnumpy.pxd%d.%d%s__builtin____builtins__214748364842949672954294967296-922337203685477580818446744073709551616__name__ndarrayflatiterbroadcast__pyx_vtable__numpy.core.multiarray_ARRAY_API_ARRAY_API not found_ARRAY_API is NULL pointermtrand.import_arrayinit mtranddtypemtrand.RandomState__getstate____setstate__permutation__%s__ returned non-%s (type %.200s)Cannot convert %.200s to %.200s%s.%s size changed, may indicate binary incompatibility%s.%s has the wrong size, try recompiling%s() got an unexpected keyword argument '%s'%s() got multiple values for keyword argument '%s''%.200s' object is unsliceablemtrand.RandomState.__getstate__mtrand.RandomState.__setstate__too many values to unpack (expected %zd)%s() takes %s %zd positional argument%s (%zd given)raise: arg 3 must be a traceback or Noneinstance exception may not have a separate valueraise: exception class must be a subclass of BaseExceptionmtrand.RandomState.permutationmtrand.RandomState.random_integersmtrand.RandomState.standard_cauchymtrand.RandomState.standard_exponentialmtrand.RandomState.standard_normalmtrand.RandomState.random_samplecan't convert negative value to unsigned charvalue too large to convert to unsigned charvalue too large to convert to signed charcan't convert negative value to unsigned intvalue too large to convert to unsigned intcan't convert negative value to unsigned shortvalue too large to convert to unsigned shortcan't convert negative value to unsigned longcan't convert negative value to unsigned PY_LONG_LONGmtrand.RandomState.standard_gammamtrand.RandomState.exponentialneed more than %zd value%s to unpackvalue too large to convert to intmtrand.RandomState.multinomialmtrand.RandomState.noncentral_chisquaremtrand.RandomState.hypergeometricmtrand.RandomState.noncentral_fmtrand.RandomState.multivariate_normalmtrand.RandomState.negative_binomialvalue too large to convert to signed shortvalue too large to convert to signed int'%.200s' object does not support slice %scompiletime version %s of module '%.100s' does not match runtime version %snumpy.core.multiarray failed to import_ARRAY_API is not PyCObject objectmodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule compiled against API version 0x%x but this version of numpy is 0x%xFATAL: module compiled as unknown endianFATAL: module compiled as little endian, but detected different endianness at runtime RandomState(seed=None) Container for the Mersenne Twister pseudo-random number generator. `RandomState` exposes a number of methods for generating random numbers drawn from a variety of probability distributions. In addition to the distribution-specific arguments, each method takes a keyword argument `size` that defaults to ``None``. If `size` is ``None``, then a single value is generated and returned. If `size` is an integer, then a 1-D array filled with generated values is returned. If `size` is a tuple, then an array with that shape is filled and returned. *Compatibility Guarantee* A fixed seed and a fixed series of calls to 'RandomState' methods using the same parameters will always produce the same results up to roundoff error except when the values were incorrect. Incorrect values will be fixed and the NumPy version in which the fix was made will be noted in the relevant docstring. Extension of existing parameter ranges and the addition of new parameters is allowed as long the previous behavior remains unchanged. Parameters ---------- seed : {None, int, array_like}, optional Random seed used to initialize the pseudo-random number generator. Can be any integer between 0 and 2**32 - 1 inclusive, an array (or other sequence) of such integers, or ``None`` (the default). If `seed` is ``None``, then `RandomState` will try to read data from ``/dev/urandom`` (or the Windows analogue) if available or seed from the clock otherwise. 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If `x` is a multi-dimensional array, it is only shuffled along its first index. Parameters ---------- x : int or array_like If `x` is an integer, randomly permute ``np.arange(x)``. If `x` is an array, make a copy and shuffle the elements randomly. Returns ------- out : ndarray Permuted sequence or array range. Examples -------- >>> np.random.permutation(10) array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) >>> np.random.permutation([1, 4, 9, 12, 15]) array([15, 1, 9, 4, 12]) >>> arr = np.arange(9).reshape((3, 3)) >>> np.random.permutation(arr) array([[6, 7, 8], [0, 1, 2], [3, 4, 5]]) shuffle(x) Modify a sequence in-place by shuffling its contents. This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same. Parameters ---------- x : array_like The array or list to be shuffled. Returns ------- None Examples -------- >>> arr = np.arange(10) >>> np.random.shuffle(arr) >>> arr [1 7 5 2 9 4 3 6 0 8] Multi-dimensional arrays are only shuffled along the first axis: >>> arr = np.arange(9).reshape((3, 3)) >>> np.random.shuffle(arr) >>> arr array([[3, 4, 5], [6, 7, 8], [0, 1, 2]]) dirichlet(alpha, size=None) Draw samples from the Dirichlet distribution. Draw `size` samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. Dirichlet pdf is the conjugate prior of a multinomial in Bayesian inference. Parameters ---------- alpha : array Parameter of the distribution (k dimension for sample of dimension k). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- samples : ndarray, The drawn samples, of shape (size, alpha.ndim). Notes ----- .. math:: X \approx \prod_{i=1}^{k}{x^{\alpha_i-1}_i} Uses the following property for computation: for each dimension, draw a random sample y_i from a standard gamma generator of shape `alpha_i`, then :math:`X = \frac{1}{\sum_{i=1}^k{y_i}} (y_1, \ldots, y_n)` is Dirichlet distributed. References ---------- .. [1] David McKay, "Information Theory, Inference and Learning Algorithms," chapter 23, http://www.inference.phy.cam.ac.uk/mackay/ .. [2] Wikipedia, "Dirichlet distribution", http://en.wikipedia.org/wiki/Dirichlet_distribution Examples -------- Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces. >>> s = np.random.dirichlet((10, 5, 3), 20).transpose() >>> plt.barh(range(20), s[0]) >>> plt.barh(range(20), s[1], left=s[0], color='g') >>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r') >>> plt.title("Lengths of Strings") multinomial(n, pvals, size=None) Draw samples from a multinomial distribution. The multinomial distribution is a multivariate generalisation of the binomial distribution. Take an experiment with one of ``p`` possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents `n` such experiments. Its values, ``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the outcome was ``i``. Parameters ---------- n : int Number of experiments. pvals : sequence of floats, length p Probabilities of each of the ``p`` different outcomes. These should sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as ``sum(pvals[:-1]) <= 1)``. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : ndarray The drawn samples, of shape *size*, if that was provided. If not, the shape is ``(N,)``. In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. Examples -------- Throw a dice 20 times: >>> np.random.multinomial(20, [1/6.]*6, size=1) array([[4, 1, 7, 5, 2, 1]]) It landed 4 times on 1, once on 2, etc. Now, throw the dice 20 times, and 20 times again: >>> np.random.multinomial(20, [1/6.]*6, size=2) array([[3, 4, 3, 3, 4, 3], [2, 4, 3, 4, 0, 7]]) For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc. A loaded die is more likely to land on number 6: >>> np.random.multinomial(100, [1/7.]*5 + [2/7.]) array([11, 16, 14, 17, 16, 26]) The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so: >>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT array([38, 62]) not like: >>> np.random.multinomial(100, [1.0, 2.0]) # WRONG array([100, 0]) multivariate_normal(mean, cov[, size, check_valid, tol]) Draw random samples from a multivariate normal distribution. The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or "center") and variance (standard deviation, or "width," squared) of the one-dimensional normal distribution. Parameters ---------- mean : 1-D array_like, of length N Mean of the N-dimensional distribution. cov : 2-D array_like, of shape (N, N) Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling. size : int or tuple of ints, optional Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are generated, and packed in an `m`-by-`n`-by-`k` arrangement. Because each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``. If no shape is specified, a single (`N`-D) sample is returned. check_valid : { 'warn', 'raise', 'ignore' }, optional Behavior when the covariance matrix is not positive semidefinite. tol : float, optional Tolerance when checking the singular values in covariance matrix. Returns ------- out : ndarray The drawn samples, of shape *size*, if that was provided. If not, the shape is ``(N,)``. In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. Notes ----- The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution. Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, :math:`X = [x_1, x_2, ... x_N]`. The covariance matrix element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`. The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its "spread"). Instead of specifying the full covariance matrix, popular approximations include: - Spherical covariance (`cov` is a multiple of the identity matrix) - Diagonal covariance (`cov` has non-negative elements, and only on the diagonal) This geometrical property can be seen in two dimensions by plotting generated data-points: >>> mean = [0, 0] >>> cov = [[1, 0], [0, 100]] # diagonal covariance Diagonal covariance means that points are oriented along x or y-axis: >>> import matplotlib.pyplot as plt >>> x, y = np.random.multivariate_normal(mean, cov, 5000).T >>> plt.plot(x, y, 'x') >>> plt.axis('equal') >>> plt.show() Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed. References ---------- .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic Processes," 3rd ed., New York: McGraw-Hill, 1991. .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern Classification," 2nd ed., New York: Wiley, 2001. Examples -------- >>> mean = (1, 2) >>> cov = [[1, 0], [0, 1]] >>> x = np.random.multivariate_normal(mean, cov, (3, 3)) >>> x.shape (3, 3, 2) The following is probably true, given that 0.6 is roughly twice the standard deviation: >>> list((x[0,0,:] - mean) < 0.6) [True, True] logseries(p, size=None) Draw samples from a logarithmic series distribution. Samples are drawn from a log series distribution with specified shape parameter, 0 < ``p`` < 1. Parameters ---------- p : float or array_like of floats Shape parameter for the distribution. Must be in the range (0, 1). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logarithmic series distribution. See Also -------- scipy.stats.logser : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Log Series distribution is .. math:: P(k) = \frac{-p^k}{k \ln(1-p)}, where p = probability. The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3]. References ---------- .. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9). .. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58. .. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994. .. [4] Wikipedia, "Logarithmic distribution", http://en.wikipedia.org/wiki/Logarithmic_distribution Examples -------- Draw samples from the distribution: >>> a = .6 >>> s = np.random.logseries(a, 10000) >>> count, bins, ignored = plt.hist(s) # plot against distribution >>> def logseries(k, p): ... return -p**k/(k*log(1-p)) >>> plt.plot(bins, logseries(bins, a)*count.max()/ logseries(bins, a).max(), 'r') >>> plt.show() hypergeometric(ngood, nbad, nsample, size=None) Draw samples from a Hypergeometric distribution. Samples are drawn from a hypergeometric distribution with specified parameters, ngood (ways to make a good selection), nbad (ways to make a bad selection), and nsample = number of items sampled, which is less than or equal to the sum ngood + nbad. Parameters ---------- ngood : int or array_like of ints Number of ways to make a good selection. Must be nonnegative. nbad : int or array_like of ints Number of ways to make a bad selection. Must be nonnegative. nsample : int or array_like of ints Number of items sampled. Must be at least 1 and at most ``ngood + nbad``. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``ngood``, ``nbad``, and ``nsample`` are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized hypergeometric distribution. See Also -------- scipy.stats.hypergeom : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Hypergeometric distribution is .. math:: P(x) = \frac{\binom{m}{n}\binom{N-m}{n-x}}{\binom{N}{n}}, where :math:`0 \le x \le m` and :math:`n+m-N \le x \le n` for P(x) the probability of x successes, n = ngood, m = nbad, and N = number of samples. Consider an urn with black and white marbles in it, ngood of them black and nbad are white. If you draw nsample balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample. Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial. References ---------- .. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972. .. [2] Weisstein, Eric W. "Hypergeometric Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html .. [3] Wikipedia, "Hypergeometric distribution", http://en.wikipedia.org/wiki/Hypergeometric_distribution Examples -------- Draw samples from the distribution: >>> ngood, nbad, nsamp = 100, 2, 10 # number of good, number of bad, and number of samples >>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000) >>> hist(s) # note that it is very unlikely to grab both bad items Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color? >>> s = np.random.hypergeometric(15, 15, 15, 100000) >>> sum(s>=12)/100000. + sum(s<=3)/100000. # answer = 0.003 ... pretty unlikely! geometric(p, size=None) Draw samples from the geometric distribution. Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, ``k = 1, 2, ...``. The probability mass function of the geometric distribution is .. math:: f(k) = (1 - p)^{k - 1} p where `p` is the probability of success of an individual trial. Parameters ---------- p : float or array_like of floats The probability of success of an individual trial. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized geometric distribution. Examples -------- Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35: >>> z = np.random.geometric(p=0.35, size=10000) How many trials succeeded after a single run? >>> (z == 1).sum() / 10000. 0.34889999999999999 #random zipf(a, size=None) Draw samples from a Zipf distribution. Samples are drawn from a Zipf distribution with specified parameter `a` > 1. The Zipf distribution (also known as the zeta distribution) is a continuous probability distribution that satisfies Zipf's law: the frequency of an item is inversely proportional to its rank in a frequency table. Parameters ---------- a : float or array_like of floats Distribution parameter. Should be greater than 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Zipf distribution. See Also -------- scipy.stats.zipf : probability density function, distribution, or cumulative density function, etc. Notes ----- The probability density for the Zipf distribution is .. math:: p(x) = \frac{x^{-a}}{\zeta(a)}, where :math:`\zeta` is the Riemann Zeta function. It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table. References ---------- .. [1] Zipf, G. K., "Selected Studies of the Principle of Relative Frequency in Language," Cambridge, MA: Harvard Univ. Press, 1932. Examples -------- Draw samples from the distribution: >>> a = 2. # parameter >>> s = np.random.zipf(a, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy import special Truncate s values at 50 so plot is interesting: >>> count, bins, ignored = plt.hist(s[s<50], 50, normed=True) >>> x = np.arange(1., 50.) >>> y = x**(-a) / special.zetac(a) >>> plt.plot(x, y/max(y), linewidth=2, color='r') >>> plt.show() poisson(lam=1.0, size=None) Draw samples from a Poisson distribution. The Poisson distribution is the limit of the binomial distribution for large N. Parameters ---------- lam : float or array_like of floats Expectation of interval, should be >= 0. A sequence of expectation intervals must be broadcastable over the requested size. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``lam`` is a scalar. Otherwise, ``np.array(lam).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Poisson distribution. Notes ----- The Poisson distribution .. math:: f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!} For events with an expected separation :math:`\lambda` the Poisson distribution :math:`f(k; \lambda)` describes the probability of :math:`k` events occurring within the observed interval :math:`\lambda`. Because the output is limited to the range of the C long type, a ValueError is raised when `lam` is within 10 sigma of the maximum representable value. References ---------- .. [1] Weisstein, Eric W. "Poisson Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html .. [2] Wikipedia, "Poisson distribution", http://en.wikipedia.org/wiki/Poisson_distribution Examples -------- Draw samples from the distribution: >>> import numpy as np >>> s = np.random.poisson(5, 10000) Display histogram of the sample: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 14, normed=True) >>> plt.show() Draw each 100 values for lambda 100 and 500: >>> s = np.random.poisson(lam=(100., 500.), size=(100, 2)) negative_binomial(n, p, size=None) Draw samples from a negative binomial distribution. Samples are drawn from a negative binomial distribution with specified parameters, `n` trials and `p` probability of success where `n` is an integer > 0 and `p` is in the interval [0, 1]. Parameters ---------- n : int or array_like of ints Parameter of the distribution, > 0. Floats are also accepted, but they will be truncated to integers. p : float or array_like of floats Parameter of the distribution, >= 0 and <=1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``n`` and ``p`` are both scalars. Otherwise, ``np.broadcast(n, p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of trials it took to achieve n - 1 successes, N - (n - 1) failures, and a success on the, (N + n)th trial. Notes ----- The probability density for the negative binomial distribution is .. math:: P(N;n,p) = \binom{N+n-1}{n-1}p^{n}(1-p)^{N}, where :math:`n-1` is the number of successes, :math:`p` is the probability of success, and :math:`N+n-1` is the number of trials. The negative binomial distribution gives the probability of n-1 successes and N failures in N+n-1 trials, and success on the (N+n)th trial. If one throws a die repeatedly until the third time a "1" appears, then the probability distribution of the number of non-"1"s that appear before the third "1" is a negative binomial distribution. References ---------- .. [1] Weisstein, Eric W. "Negative Binomial Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html .. [2] Wikipedia, "Negative binomial distribution", http://en.wikipedia.org/wiki/Negative_binomial_distribution Examples -------- Draw samples from the distribution: A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.? >>> s = np.random.negative_binomial(1, 0.1, 100000) >>> for i in range(1, 11): ... probability = sum(s>> n, p = 10, .5 # number of trials, probability of each trial >>> s = np.random.binomial(n, p, 1000) # result of flipping a coin 10 times, tested 1000 times. A real world example. A company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of 0.1. All nine wells fail. What is the probability of that happening? Let's do 20,000 trials of the model, and count the number that generate zero positive results. >>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000. # answer = 0.38885, or 38%. triangular(left, mode, right, size=None) Draw samples from the triangular distribution over the interval ``[left, right]``. The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf. Parameters ---------- left : float or array_like of floats Lower limit. mode : float or array_like of floats The value where the peak of the distribution occurs. The value should fulfill the condition ``left <= mode <= right``. right : float or array_like of floats Upper limit, should be larger than `left`. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``left``, ``mode``, and ``right`` are all scalars. Otherwise, ``np.broadcast(left, mode, right).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized triangular distribution. Notes ----- The probability density function for the triangular distribution is .. math:: P(x;l, m, r) = \begin{cases} \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases} The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations. References ---------- .. [1] Wikipedia, "Triangular distribution" http://en.wikipedia.org/wiki/Triangular_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200, ... normed=True) >>> plt.show() wald(mean, scale, size=None) Draw samples from a Wald, or inverse Gaussian, distribution. As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal. The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time. Parameters ---------- mean : float or array_like of floats Distribution mean, should be > 0. scale : float or array_like of floats Scale parameter, should be >= 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(mean, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Wald distribution. Notes ----- The probability density function for the Wald distribution is .. math:: P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x} As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes. References ---------- .. [1] Brighton Webs Ltd., Wald Distribution, http://www.brighton-webs.co.uk/distributions/wald.asp .. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian Distribution: Theory : Methodology, and Applications", CRC Press, 1988. .. [3] Wikipedia, "Wald distribution" http://en.wikipedia.org/wiki/Wald_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, normed=True) >>> plt.show() rayleigh(scale=1.0, size=None) Draw samples from a Rayleigh distribution. The :math:`\chi` and Weibull distributions are generalizations of the Rayleigh. Parameters ---------- scale : float or array_like of floats, optional Scale, also equals the mode. Should be >= 0. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Rayleigh distribution. Notes ----- The probability density function for the Rayleigh distribution is .. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}} The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution. References ---------- .. [1] Brighton Webs Ltd., "Rayleigh Distribution," http://www.brighton-webs.co.uk/distributions/rayleigh.asp .. [2] Wikipedia, "Rayleigh distribution" http://en.wikipedia.org/wiki/Rayleigh_distribution Examples -------- Draw values from the distribution and plot the histogram >>> values = hist(np.random.rayleigh(3, 100000), bins=200, normed=True) Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters? >>> meanvalue = 1 >>> modevalue = np.sqrt(2 / np.pi) * meanvalue >>> s = np.random.rayleigh(modevalue, 1000000) The percentage of waves larger than 3 meters is: >>> 100.*sum(s>3)/1000000. 0.087300000000000003 lognormal(mean=0.0, sigma=1.0, size=None) Draw samples from a log-normal distribution. Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from. Parameters ---------- mean : float or array_like of floats, optional Mean value of the underlying normal distribution. Default is 0. sigma : float or array_like of floats, optional Standard deviation of the underlying normal distribution. Should be greater than zero. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``sigma`` are both scalars. Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized log-normal distribution. See Also -------- scipy.stats.lognorm : probability density function, distribution, cumulative density function, etc. Notes ----- A variable `x` has a log-normal distribution if `log(x)` is normally distributed. The probability density function for the log-normal distribution is: .. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})} where :math:`\mu` is the mean and :math:`\sigma` is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the *product* of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the *sum* of a large number of independent, identically-distributed variables. References ---------- .. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal Distributions across the Sciences: Keys and Clues," BioScience, Vol. 51, No. 5, May, 2001. http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf .. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme Values," Basel: Birkhauser Verlag, 2001, pp. 31-32. Examples -------- Draw samples from the distribution: >>> mu, sigma = 3., 1. # mean and standard deviation >>> s = np.random.lognormal(mu, sigma, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 100, normed=True, align='mid') >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, linewidth=2, color='r') >>> plt.axis('tight') >>> plt.show() Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function. >>> # Generate a thousand samples: each is the product of 100 random >>> # values, drawn from a normal distribution. >>> b = [] >>> for i in range(1000): ... a = 10. + np.random.random(100) ... b.append(np.product(a)) >>> b = np.array(b) / np.min(b) # scale values to be positive >>> count, bins, ignored = plt.hist(b, 100, normed=True, align='mid') >>> sigma = np.std(np.log(b)) >>> mu = np.mean(np.log(b)) >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, color='r', linewidth=2) >>> plt.show() logistic(loc=0.0, scale=1.0, size=None) Draw samples from a logistic distribution. Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0). Parameters ---------- loc : float or array_like of floats, optional Parameter of the distribution. Default is 0. scale : float or array_like of floats, optional Parameter of the distribution. Should be greater than zero. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logistic distribution. See Also -------- scipy.stats.logistic : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Logistic distribution is .. math:: P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2}, where :math:`\mu` = location and :math:`s` = scale. The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable. References ---------- .. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields," Birkhauser Verlag, Basel, pp 132-133. .. [2] Weisstein, Eric W. "Logistic Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html .. [3] Wikipedia, "Logistic-distribution", http://en.wikipedia.org/wiki/Logistic_distribution Examples -------- Draw samples from the distribution: >>> loc, scale = 10, 1 >>> s = np.random.logistic(loc, scale, 10000) >>> count, bins, ignored = plt.hist(s, bins=50) # plot against distribution >>> def logist(x, loc, scale): ... return exp((loc-x)/scale)/(scale*(1+exp((loc-x)/scale))**2) >>> plt.plot(bins, logist(bins, loc, scale)*count.max()/\ ... logist(bins, loc, scale).max()) >>> plt.show() gumbel(loc=0.0, scale=1.0, size=None) Draw samples from a Gumbel distribution. Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below. Parameters ---------- loc : float or array_like of floats, optional The location of the mode of the distribution. Default is 0. scale : float or array_like of floats, optional The scale parameter of the distribution. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Gumbel distribution. See Also -------- scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull Notes ----- The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with "exponential-like" tails. The probability density for the Gumbel distribution is .. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}}, where :math:`\mu` is the mode, a location parameter, and :math:`\beta` is the scale parameter. The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a "fat-tailed" distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events. It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet. The function has a mean of :math:`\mu + 0.57721\beta` and a variance of :math:`\frac{\pi^2}{6}\beta^2`. References ---------- .. [1] Gumbel, E. J., "Statistics of Extremes," New York: Columbia University Press, 1958. .. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields," Basel: Birkhauser Verlag, 2001. Examples -------- Draw samples from the distribution: >>> mu, beta = 0, 0.1 # location and scale >>> s = np.random.gumbel(mu, beta, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, normed=True) >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp( -np.exp( -(bins - mu) /beta) ), ... linewidth=2, color='r') >>> plt.show() Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian: >>> means = [] >>> maxima = [] >>> for i in range(0,1000) : ... a = np.random.normal(mu, beta, 1000) ... means.append(a.mean()) ... maxima.append(a.max()) >>> count, bins, ignored = plt.hist(maxima, 30, normed=True) >>> beta = np.std(maxima) * np.sqrt(6) / np.pi >>> mu = np.mean(maxima) - 0.57721*beta >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp(-np.exp(-(bins - mu)/beta)), ... linewidth=2, color='r') >>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi)) ... * np.exp(-(bins - mu)**2 / (2 * beta**2)), ... linewidth=2, color='g') >>> plt.show() laplace(loc=0.0, scale=1.0, size=None) Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay). The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables. Parameters ---------- loc : float or array_like of floats, optional The position, :math:`\mu`, of the distribution peak. Default is 0. scale : float or array_like of floats, optional :math:`\lambda`, the exponential decay. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Laplace distribution. Notes ----- It has the probability density function .. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right). The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] Kotz, Samuel, et. al. "The Laplace Distribution and Generalizations, " Birkhauser, 2001. .. [3] Weisstein, Eric W. "Laplace Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html .. [4] Wikipedia, "Laplace distribution", http://en.wikipedia.org/wiki/Laplace_distribution Examples -------- Draw samples from the distribution >>> loc, scale = 0., 1. >>> s = np.random.laplace(loc, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, normed=True) >>> x = np.arange(-8., 8., .01) >>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale) >>> plt.plot(x, pdf) Plot Gaussian for comparison: >>> g = (1/(scale * np.sqrt(2 * np.pi)) * ... np.exp(-(x - loc)**2 / (2 * scale**2))) >>> plt.plot(x,g) power(a, size=None) Draws samples in [0, 1] from a power distribution with positive exponent a - 1. Also known as the power function distribution. Parameters ---------- a : float or array_like of floats Parameter of the distribution. Should be greater than zero. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized power distribution. Raises ------ ValueError If a < 1. Notes ----- The probability density function is .. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0. The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution. It is used, for example, in modeling the over-reporting of insurance claims. References ---------- .. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions in economics and actuarial sciences", Wiley, 2003. .. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions", National Institute of Standards and Technology Handbook Series, June 2003. http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf Examples -------- Draw samples from the distribution: >>> a = 5. # shape >>> samples = 1000 >>> s = np.random.power(a, samples) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, bins=30) >>> x = np.linspace(0, 1, 100) >>> y = a*x**(a-1.) >>> normed_y = samples*np.diff(bins)[0]*y >>> plt.plot(x, normed_y) >>> plt.show() Compare the power function distribution to the inverse of the Pareto. >>> from scipy import stats >>> rvs = np.random.power(5, 1000000) >>> rvsp = np.random.pareto(5, 1000000) >>> xx = np.linspace(0,1,100) >>> powpdf = stats.powerlaw.pdf(xx,5) >>> plt.figure() >>> plt.hist(rvs, bins=50, normed=True) >>> plt.plot(xx,powpdf,'r-') >>> plt.title('np.random.power(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, normed=True) >>> plt.plot(xx,powpdf,'r-') >>> plt.title('inverse of 1 + np.random.pareto(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, normed=True) >>> plt.plot(xx,powpdf,'r-') >>> plt.title('inverse of stats.pareto(5)') weibull(a, size=None) Draw samples from a Weibull distribution. Draw samples from a 1-parameter Weibull distribution with the given shape parameter `a`. .. math:: X = (-ln(U))^{1/a} Here, U is drawn from the uniform distribution over (0,1]. The more common 2-parameter Weibull, including a scale parameter :math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`. Parameters ---------- a : float or array_like of floats Shape of the distribution. Should be greater than zero. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Weibull distribution. See Also -------- scipy.stats.weibull_max scipy.stats.weibull_min scipy.stats.genextreme gumbel Notes ----- The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions. The probability density for the Weibull distribution is .. math:: p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a}, where :math:`a` is the shape and :math:`\lambda` the scale. The function has its peak (the mode) at :math:`\lambda(\frac{a-1}{a})^{1/a}`. When ``a = 1``, the Weibull distribution reduces to the exponential distribution. References ---------- .. [1] Waloddi Weibull, Royal Technical University, Stockholm, 1939 "A Statistical Theory Of The Strength Of Materials", Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm. .. [2] Waloddi Weibull, "A Statistical Distribution Function of Wide Applicability", Journal Of Applied Mechanics ASME Paper 1951. .. [3] Wikipedia, "Weibull distribution", http://en.wikipedia.org/wiki/Weibull_distribution Examples -------- Draw samples from the distribution: >>> a = 5. # shape >>> s = np.random.weibull(a, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> x = np.arange(1,100.)/50. >>> def weib(x,n,a): ... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a) >>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000)) >>> x = np.arange(1,100.)/50. >>> scale = count.max()/weib(x, 1., 5.).max() >>> plt.plot(x, weib(x, 1., 5.)*scale) >>> plt.show() pareto(a, size=None) Draw samples from a Pareto II or Lomax distribution with specified shape. The Lomax or Pareto II distribution is a shifted Pareto distribution. The classical Pareto distribution can be obtained from the Lomax distribution by adding 1 and multiplying by the scale parameter ``m`` (see Notes). The smallest value of the Lomax distribution is zero while for the classical Pareto distribution it is ``mu``, where the standard Pareto distribution has location ``mu = 1``. Lomax can also be considered as a simplified version of the Generalized Pareto distribution (available in SciPy), with the scale set to one and the location set to zero. The Pareto distribution must be greater than zero, and is unbounded above. It is also known as the "80-20 rule". In this distribution, 80 percent of the weights are in the lowest 20 percent of the range, while the other 20 percent fill the remaining 80 percent of the range. Parameters ---------- a : float or array_like of floats Shape of the distribution. Should be greater than zero. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Pareto distribution. See Also -------- scipy.stats.lomax : probability density function, distribution or cumulative density function, etc. scipy.stats.genpareto : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Pareto distribution is .. math:: p(x) = \frac{am^a}{x^{a+1}} where :math:`a` is the shape and :math:`m` the scale. The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution useful in many real world problems. Outside the field of economics it is generally referred to as the Bradford distribution. Pareto developed the distribution to describe the distribution of wealth in an economy. It has also found use in insurance, web page access statistics, oil field sizes, and many other problems, including the download frequency for projects in Sourceforge [1]_. It is one of the so-called "fat-tailed" distributions. References ---------- .. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects. .. [2] Pareto, V. (1896). Course of Political Economy. Lausanne. .. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30. .. [4] Wikipedia, "Pareto distribution", http://en.wikipedia.org/wiki/Pareto_distribution Examples -------- Draw samples from the distribution: >>> a, m = 3., 2. # shape and mode >>> s = (np.random.pareto(a, 1000) + 1) * m Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 100, normed=True) >>> fit = a*m**a / bins**(a+1) >>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r') >>> plt.show() vonmises(mu, kappa, size=None) Draw samples from a von Mises distribution. Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi]. The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution. Parameters ---------- mu : float or array_like of floats Mode ("center") of the distribution. kappa : float or array_like of floats Dispersion of the distribution, has to be >=0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mu`` and ``kappa`` are both scalars. Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized von Mises distribution. See Also -------- scipy.stats.vonmises : probability density function, distribution, or cumulative density function, etc. Notes ----- The probability density for the von Mises distribution is .. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}, where :math:`\mu` is the mode and :math:`\kappa` the dispersion, and :math:`I_0(\kappa)` is the modified Bessel function of order 0. The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] von Mises, R., "Mathematical Theory of Probability and Statistics", New York: Academic Press, 1964. Examples -------- Draw samples from the distribution: >>> mu, kappa = 0.0, 4.0 # mean and dispersion >>> s = np.random.vonmises(mu, kappa, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import i0 >>> plt.hist(s, 50, normed=True) >>> x = np.linspace(-np.pi, np.pi, num=51) >>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa)) >>> plt.plot(x, y, linewidth=2, color='r') >>> plt.show() standard_t(df, size=None) Draw samples from a standard Student's t distribution with `df` degrees of freedom. A special case of the hyperbolic distribution. As `df` gets large, the result resembles that of the standard normal distribution (`standard_normal`). Parameters ---------- df : int or array_like of ints Degrees of freedom, should be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard Student's t distribution. Notes ----- The probability density function for the t distribution is .. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2} The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean. The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student. References ---------- .. [1] Dalgaard, Peter, "Introductory Statistics With R", Springer, 2002. .. [2] Wikipedia, "Student's t-distribution" http://en.wikipedia.org/wiki/Student's_t-distribution Examples -------- From Dalgaard page 83 [1]_, suppose the daily energy intake for 11 women in Kj is: >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \ ... 7515, 8230, 8770]) Does their energy intake deviate systematically from the recommended value of 7725 kJ? We have 10 degrees of freedom, so is the sample mean within 95% of the recommended value? >>> s = np.random.standard_t(10, size=100000) >>> np.mean(intake) 6753.636363636364 >>> intake.std(ddof=1) 1142.1232221373727 Calculate the t statistic, setting the ddof parameter to the unbiased value so the divisor in the standard deviation will be degrees of freedom, N-1. >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake))) >>> import matplotlib.pyplot as plt >>> h = plt.hist(s, bins=100, normed=True) For a one-sided t-test, how far out in the distribution does the t statistic appear? >>> np.sum(s>> s = np.random.standard_cauchy(1000000) >>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well >>> plt.hist(s, bins=100) >>> plt.show() noncentral_chisquare(df, nonc, size=None) Draw samples from a noncentral chi-square distribution. The noncentral :math:`\chi^2` distribution is a generalisation of the :math:`\chi^2` distribution. Parameters ---------- df : int or array_like of ints Degrees of freedom, should be > 0 as of NumPy 1.10.0, should be > 1 for earlier versions. nonc : float or array_like of floats Non-centrality, should be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` and ``nonc`` are both scalars. Otherwise, ``np.broadcast(df, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral chi-square distribution. Notes ----- The probability density function for the noncentral Chi-square distribution is .. math:: P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} \P_{Y_{df+2i}}(x), where :math:`Y_{q}` is the Chi-square with q degrees of freedom. In Delhi (2007), it is noted that the noncentral chi-square is useful in bombing and coverage problems, the probability of killing the point target given by the noncentral chi-squared distribution. References ---------- .. [1] Delhi, M.S. Holla, "On a noncentral chi-square distribution in the analysis of weapon systems effectiveness", Metrika, Volume 15, Number 1 / December, 1970. .. [2] Wikipedia, "Noncentral chi-square distribution" http://en.wikipedia.org/wiki/Noncentral_chi-square_distribution Examples -------- Draw values from the distribution and plot the histogram >>> import matplotlib.pyplot as plt >>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000), ... bins=200, normed=True) >>> plt.show() Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare. >>> plt.figure() >>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000), ... bins=np.arange(0., 25, .1), normed=True) >>> values2 = plt.hist(np.random.chisquare(3, 100000), ... bins=np.arange(0., 25, .1), normed=True) >>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob') >>> plt.show() Demonstrate how large values of non-centrality lead to a more symmetric distribution. >>> plt.figure() >>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000), ... bins=200, normed=True) >>> plt.show() chisquare(df, size=None) Draw samples from a chi-square distribution. When `df` independent random variables, each with standard normal distributions (mean 0, variance 1), are squared and summed, the resulting distribution is chi-square (see Notes). This distribution is often used in hypothesis testing. Parameters ---------- df : int or array_like of ints Number of degrees of freedom. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized chi-square distribution. Raises ------ ValueError When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``) is given. Notes ----- The variable obtained by summing the squares of `df` independent, standard normally distributed random variables: .. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i is chi-square distributed, denoted .. math:: Q \sim \chi^2_k. The probability density function of the chi-squared distribution is .. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, where :math:`\Gamma` is the gamma function, .. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt. References ---------- .. [1] NIST "Engineering Statistics Handbook" http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Examples -------- >>> np.random.chisquare(2,4) array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) noncentral_f(dfnum, dfden, nonc, size=None) Draw samples from the noncentral F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters > 1. `nonc` is the non-centrality parameter. Parameters ---------- dfnum : int or array_like of ints Parameter, should be > 1. dfden : int or array_like of ints Parameter, should be > 1. nonc : float or array_like of floats Parameter, should be >= 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum``, ``dfden``, and ``nonc`` are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral Fisher distribution. Notes ----- When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic. References ---------- .. [1] Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html .. [2] Wikipedia, "Noncentral F-distribution", http://en.wikipedia.org/wiki/Noncentral_F-distribution Examples -------- In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We'll plot the two probability distributions for comparison. >>> dfnum = 3 # between group deg of freedom >>> dfden = 20 # within groups degrees of freedom >>> nonc = 3.0 >>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000) >>> NF = np.histogram(nc_vals, bins=50, normed=True) >>> c_vals = np.random.f(dfnum, dfden, 1000000) >>> F = np.histogram(c_vals, bins=50, normed=True) >>> plt.plot(F[1][1:], F[0]) >>> plt.plot(NF[1][1:], NF[0]) >>> plt.show() f(dfnum, dfden, size=None) Draw samples from an F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters should be greater than zero. The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates. Parameters ---------- dfnum : int or array_like of ints Degrees of freedom in numerator. Should be greater than zero. dfden : int or array_like of ints Degrees of freedom in denominator. Should be greater than zero. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum`` and ``dfden`` are both scalars. Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Fisher distribution. See Also -------- scipy.stats.f : probability density function, distribution or cumulative density function, etc. Notes ----- The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable `dfnum` is the number of samples minus one, the between-groups degrees of freedom, while `dfden` is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups. References ---------- .. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002. .. [2] Wikipedia, "F-distribution", http://en.wikipedia.org/wiki/F-distribution Examples -------- An example from Glantz[1], pp 47-40: Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children's blood glucose levels? Calculating the F statistic from the data gives a value of 36.01. Draw samples from the distribution: >>> dfnum = 1. # between group degrees of freedom >>> dfden = 48. # within groups degrees of freedom >>> s = np.random.f(dfnum, dfden, 1000) The lower bound for the top 1% of the samples is : >>> sort(s)[-10] 7.61988120985 So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level. gamma(shape, scale=1.0, size=None) Draw samples from a Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, `shape` (sometimes designated "k") and `scale` (sometimes designated "theta"), where both parameters are > 0. Parameters ---------- shape : float or array_like of floats The shape of the gamma distribution. Should be greater than zero. scale : float or array_like of floats, optional The scale of the gamma distribution. Should be greater than zero. Default is equal to 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", http://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2) >>> s = np.random.gamma(shape, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps >>> count, bins, ignored = plt.hist(s, 50, normed=True) >>> y = bins**(shape-1)*(np.exp(-bins/scale) / ... (sps.gamma(shape)*scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') >>> plt.show() standard_gamma(shape, size=None) Draw samples from a standard Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated "k") and scale=1. Parameters ---------- shape : float or array_like of floats Parameter, should be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` is a scalar. Otherwise, ``np.array(shape).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", http://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 1. # mean and width >>> s = np.random.standard_gamma(shape, 1000000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps >>> count, bins, ignored = plt.hist(s, 50, normed=True) >>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ \ ... (sps.gamma(shape) * scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') >>> plt.show() standard_exponential(size=None) Draw samples from the standard exponential distribution. `standard_exponential` is identical to the exponential distribution with a scale parameter of 1. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray Drawn samples. Examples -------- Output a 3x8000 array: >>> n = np.random.standard_exponential((3, 8000)) exponential(scale=1.0, size=None) Draw samples from an exponential distribution. Its probability density function is .. math:: f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}), for ``x > 0`` and 0 elsewhere. :math:`\beta` is the scale parameter, which is the inverse of the rate parameter :math:`\lambda = 1/\beta`. The rate parameter is an alternative, widely used parameterization of the exponential distribution [3]_. The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms [1]_, or the time between page requests to Wikipedia [2]_. Parameters ---------- scale : float or array_like of floats The scale parameter, :math:`\beta = 1/\lambda`. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized exponential distribution. References ---------- .. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and Random Signal Principles", 4th ed, 2001, p. 57. .. [2] Wikipedia, "Poisson process", http://en.wikipedia.org/wiki/Poisson_process .. [3] Wikipedia, "Exponential distribution", http://en.wikipedia.org/wiki/Exponential_distribution beta(a, b, size=None) Draw samples from a Beta distribution. The Beta distribution is a special case of the Dirichlet distribution, and is related to the Gamma distribution. It has the probability distribution function .. math:: f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}, where the normalisation, B, is the beta function, .. math:: B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt. It is often seen in Bayesian inference and order statistics. Parameters ---------- a : float or array_like of floats Alpha, non-negative. b : float or array_like of floats Beta, non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` and ``b`` are both scalars. Otherwise, ``np.broadcast(a, b).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized beta distribution. normal(loc=0.0, scale=1.0, size=None) Draw random samples from a normal (Gaussian) distribution. The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2]_, is often called the bell curve because of its characteristic shape (see the example below). The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2]_. Parameters ---------- loc : float or array_like of floats Mean ("centre") of the distribution. scale : float or array_like of floats Standard deviation (spread or "width") of the distribution. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized normal distribution. See Also -------- scipy.stats.norm : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gaussian distribution is .. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} }, where :math:`\mu` is the mean and :math:`\sigma` the standard deviation. The square of the standard deviation, :math:`\sigma^2`, is called the variance. The function has its peak at the mean, and its "spread" increases with the standard deviation (the function reaches 0.607 times its maximum at :math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that `numpy.random.normal` is more likely to return samples lying close to the mean, rather than those far away. References ---------- .. [1] Wikipedia, "Normal distribution", http://en.wikipedia.org/wiki/Normal_distribution .. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability, Random Variables and Random Signal Principles", 4th ed., 2001, pp. 51, 51, 125. Examples -------- Draw samples from the distribution: >>> mu, sigma = 0, 0.1 # mean and standard deviation >>> s = np.random.normal(mu, sigma, 1000) Verify the mean and the variance: >>> abs(mu - np.mean(s)) < 0.01 True >>> abs(sigma - np.std(s, ddof=1)) < 0.01 True Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, normed=True) >>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) * ... np.exp( - (bins - mu)**2 / (2 * sigma**2) ), ... linewidth=2, color='r') >>> plt.show() standard_normal(size=None) Draw samples from a standard Normal distribution (mean=0, stdev=1). Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray Drawn samples. Examples -------- >>> s = np.random.standard_normal(8000) >>> s array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, #random -0.38672696, -0.4685006 ]) #random >>> s.shape (8000,) >>> s = np.random.standard_normal(size=(3, 4, 2)) >>> s.shape (3, 4, 2) random_integers(low, high=None, size=None) Random integers of type np.int between `low` and `high`, inclusive. Return random integers of type np.int from the "discrete uniform" distribution in the closed interval [`low`, `high`]. If `high` is None (the default), then results are from [1, `low`]. The np.int type translates to the C long type used by Python 2 for "short" integers and its precision is platform dependent. This function has been deprecated. Use randint instead. .. deprecated:: 1.11.0 Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution (unless ``high=None``, in which case this parameter is the *highest* such integer). high : int, optional If provided, the largest (signed) integer to be drawn from the distribution (see above for behavior if ``high=None``). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : int or ndarray of ints `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. See Also -------- random.randint : Similar to `random_integers`, only for the half-open interval [`low`, `high`), and 0 is the lowest value if `high` is omitted. Notes ----- To sample from N evenly spaced floating-point numbers between a and b, use:: a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.) Examples -------- >>> np.random.random_integers(5) 4 >>> type(np.random.random_integers(5)) >>> np.random.random_integers(5, size=(3,2)) array([[5, 4], [3, 3], [4, 5]]) Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (*i.e.*, from the set :math:`{0, 5/8, 10/8, 15/8, 20/8}`): >>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4. array([ 0.625, 1.25 , 0.625, 0.625, 2.5 ]) Roll two six sided dice 1000 times and sum the results: >>> d1 = np.random.random_integers(1, 6, 1000) >>> d2 = np.random.random_integers(1, 6, 1000) >>> dsums = d1 + d2 Display results as a histogram: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(dsums, 11, normed=True) >>> plt.show() randn(d0, d1, ..., dn) Return a sample (or samples) from the "standard normal" distribution. If positive, int_like or int-convertible arguments are provided, `randn` generates an array of shape ``(d0, d1, ..., dn)``, filled with random floats sampled from a univariate "normal" (Gaussian) distribution of mean 0 and variance 1 (if any of the :math:`d_i` are floats, they are first converted to integers by truncation). A single float randomly sampled from the distribution is returned if no argument is provided. This is a convenience function. If you want an interface that takes a tuple as the first argument, use `numpy.random.standard_normal` instead. Parameters ---------- d0, d1, ..., dn : int, optional The dimensions of the returned array, should be all positive. If no argument is given a single Python float is returned. Returns ------- Z : ndarray or float A ``(d0, d1, ..., dn)``-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied. See Also -------- random.standard_normal : Similar, but takes a tuple as its argument. Notes ----- For random samples from :math:`N(\mu, \sigma^2)`, use: ``sigma * np.random.randn(...) + mu`` Examples -------- >>> np.random.randn() 2.1923875335537315 #random Two-by-four array of samples from N(3, 6.25): >>> 2.5 * np.random.randn(2, 4) + 3 array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], #random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) #random rand(d0, d1, ..., dn) Random values in a given shape. Create an array of the given shape and populate it with random samples from a uniform distribution over ``[0, 1)``. Parameters ---------- d0, d1, ..., dn : int, optional The dimensions of the returned array, should all be positive. If no argument is given a single Python float is returned. Returns ------- out : ndarray, shape ``(d0, d1, ..., dn)`` Random values. See Also -------- random Notes ----- This is a convenience function. If you want an interface that takes a shape-tuple as the first argument, refer to np.random.random_sample . Examples -------- >>> np.random.rand(3,2) array([[ 0.14022471, 0.96360618], #random [ 0.37601032, 0.25528411], #random [ 0.49313049, 0.94909878]]) #random uniform(low=0.0, high=1.0, size=None) Draw samples from a uniform distribution. Samples are uniformly distributed over the half-open interval ``[low, high)`` (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by `uniform`. Parameters ---------- low : float or array_like of floats, optional Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0. high : float or array_like of floats Upper boundary of the output interval. All values generated will be less than high. The default value is 1.0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``low`` and ``high`` are both scalars. Otherwise, ``np.broadcast(low, high).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized uniform distribution. See Also -------- randint : Discrete uniform distribution, yielding integers. random_integers : Discrete uniform distribution over the closed interval ``[low, high]``. random_sample : Floats uniformly distributed over ``[0, 1)``. random : Alias for `random_sample`. rand : Convenience function that accepts dimensions as input, e.g., ``rand(2,2)`` would generate a 2-by-2 array of floats, uniformly distributed over ``[0, 1)``. Notes ----- The probability density function of the uniform distribution is .. math:: p(x) = \frac{1}{b - a} anywhere within the interval ``[a, b)``, and zero elsewhere. When ``high`` == ``low``, values of ``low`` will be returned. If ``high`` < ``low``, the results are officially undefined and may eventually raise an error, i.e. do not rely on this function to behave when passed arguments satisfying that inequality condition. Examples -------- Draw samples from the distribution: >>> s = np.random.uniform(-1,0,1000) All values are within the given interval: >>> np.all(s >= -1) True >>> np.all(s < 0) True Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 15, normed=True) >>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r') >>> plt.show() choice(a, size=None, replace=True, p=None) Generates a random sample from a given 1-D array .. versionadded:: 1.7.0 Parameters ----------- a : 1-D array-like or int If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if a were np.arange(a) size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. replace : boolean, optional Whether the sample is with or without replacement p : 1-D array-like, optional The probabilities associated with each entry in a. If not given the sample assumes a uniform distribution over all entries in a. Returns -------- samples : single item or ndarray The generated random samples Raises ------- ValueError If a is an int and less than zero, if a or p are not 1-dimensional, if a is an array-like of size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size See Also --------- randint, shuffle, permutation Examples --------- Generate a uniform random sample from np.arange(5) of size 3: >>> np.random.choice(5, 3) array([0, 3, 4]) >>> #This is equivalent to np.random.randint(0,5,3) Generate a non-uniform random sample from np.arange(5) of size 3: >>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0]) array([3, 3, 0]) Generate a uniform random sample from np.arange(5) of size 3 without replacement: >>> np.random.choice(5, 3, replace=False) array([3,1,0]) >>> #This is equivalent to np.random.permutation(np.arange(5))[:3] Generate a non-uniform random sample from np.arange(5) of size 3 without replacement: >>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0]) array([2, 3, 0]) Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance: >>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher'] >>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3]) array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], dtype='|S11') bytes(length) Return random bytes. Parameters ---------- length : int Number of random bytes. Returns ------- out : str String of length `length`. Examples -------- >>> np.random.bytes(10) ' eh\x85\x022SZ\xbf\xa4' #random randint(low, high=None, size=None, dtype='l') Return random integers from `low` (inclusive) to `high` (exclusive). Return random integers from the "discrete uniform" distribution of the specified dtype in the "half-open" interval [`low`, `high`). If `high` is None (the default), then results are from [0, `low`). Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution (unless ``high=None``, in which case this parameter is one above the *highest* such integer). high : int, optional If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if ``high=None``). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result. All dtypes are determined by their name, i.e., 'int64', 'int', etc, so byteorder is not available and a specific precision may have different C types depending on the platform. The default value is 'np.int'. .. versionadded:: 1.11.0 Returns ------- out : int or ndarray of ints `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. See Also -------- random.random_integers : similar to `randint`, only for the closed interval [`low`, `high`], and 1 is the lowest value if `high` is omitted. In particular, this other one is the one to use to generate uniformly distributed discrete non-integers. Examples -------- >>> np.random.randint(2, size=10) array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) >>> np.random.randint(1, size=10) array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) Generate a 2 x 4 array of ints between 0 and 4, inclusive: >>> np.random.randint(5, size=(2, 4)) array([[4, 0, 2, 1], [3, 2, 2, 0]]) tomaxint(size=None) Random integers between 0 and ``sys.maxint``, inclusive. Return a sample of uniformly distributed random integers in the interval [0, ``sys.maxint``]. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : ndarray Drawn samples, with shape `size`. See Also -------- randint : Uniform sampling over a given half-open interval of integers. random_integers : Uniform sampling over a given closed interval of integers. Examples -------- >>> RS = np.random.mtrand.RandomState() # need a RandomState object >>> RS.tomaxint((2,2,2)) array([[[1170048599, 1600360186], [ 739731006, 1947757578]], [[1871712945, 752307660], [1601631370, 1479324245]]]) >>> import sys >>> sys.maxint 2147483647 >>> RS.tomaxint((2,2,2)) < sys.maxint array([[[ True, True], [ True, True]], [[ True, True], [ True, True]]], dtype=bool) random_sample(size=None) Return random floats in the half-open interval [0.0, 1.0). Results are from the "continuous uniform" distribution over the stated interval. To sample :math:`Unif[a, b), b > a` multiply the output of `random_sample` by `(b-a)` and add `a`:: (b - a) * random_sample() + a Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray of floats Array of random floats of shape `size` (unless ``size=None``, in which case a single float is returned). Examples -------- >>> np.random.random_sample() 0.47108547995356098 >>> type(np.random.random_sample()) >>> np.random.random_sample((5,)) array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428]) Three-by-two array of random numbers from [-5, 0): >>> 5 * np.random.random_sample((3, 2)) - 5 array([[-3.99149989, -0.52338984], [-2.99091858, -0.79479508], [-1.23204345, -1.75224494]]) set_state(state) Set the internal state of the generator from a tuple. For use if one has reason to manually (re-)set the internal state of the "Mersenne Twister"[1]_ pseudo-random number generating algorithm. Parameters ---------- state : tuple(str, ndarray of 624 uints, int, int, float) The `state` tuple has the following items: 1. the string 'MT19937', specifying the Mersenne Twister algorithm. 2. a 1-D array of 624 unsigned integers ``keys``. 3. an integer ``pos``. 4. an integer ``has_gauss``. 5. a float ``cached_gaussian``. Returns ------- out : None Returns 'None' on success. See Also -------- get_state Notes ----- `set_state` and `get_state` are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing. For backwards compatibility, the form (str, array of 624 uints, int) is also accepted although it is missing some information about the cached Gaussian value: ``state = ('MT19937', keys, pos)``. References ---------- .. [1] M. Matsumoto and T. Nishimura, "Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator," *ACM Trans. on Modeling and Computer Simulation*, Vol. 8, No. 1, pp. 3-30, Jan. 1998. get_state() Return a tuple representing the internal state of the generator. For more details, see `set_state`. Returns ------- out : tuple(str, ndarray of 624 uints, int, int, float) The returned tuple has the following items: 1. the string 'MT19937'. 2. a 1-D array of 624 unsigned integer keys. 3. an integer ``pos``. 4. an integer ``has_gauss``. 5. a float ``cached_gaussian``. See Also -------- set_state Notes ----- `set_state` and `get_state` are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing. seed(seed=None) Seed the generator. This method is called when `RandomState` is initialized. It can be called again to re-seed the generator. For details, see `RandomState`. Parameters ---------- seed : int or array_like, optional Seed for `RandomState`. Must be convertible to 32 bit unsigned integers. See Also -------- RandomState _rand_uint64(low, high, size, rngstate) Return random np.uint64 integers between ``low`` and ``high``, inclusive. Return random integers from the "discrete uniform" distribution in the closed interval [``low``, ``high``). On entry the arguments are presumed to have been validated for size and order for the np.uint64 type. Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution. high : int Highest (signed) integer to be drawn from the distribution. size : int or tuple of ints Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. rngstate : encapsulated pointer to rk_state The specific type depends on the python version. In Python 2 it is a PyCObject, in Python 3 a PyCapsule object. Returns ------- out : python integer or ndarray of np.uint64 `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. _rand_uint32(low, high, size, rngstate) Return random np.uint32 integers between ``low`` and ``high``, inclusive. Return random integers from the "discrete uniform" distribution in the closed interval [``low``, ``high``). On entry the arguments are presumed to have been validated for size and order for the np.uint32 type. Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution. high : int Highest (signed) integer to be drawn from the distribution. size : int or tuple of ints Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. rngstate : encapsulated pointer to rk_state The specific type depends on the python version. In Python 2 it is a PyCObject, in Python 3 a PyCapsule object. Returns ------- out : python integer or ndarray of np.uint32 `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. _rand_uint16(low, high, size, rngstate) Return random np.uint16 integers between ``low`` and ``high``, inclusive. Return random integers from the "discrete uniform" distribution in the closed interval [``low``, ``high``). On entry the arguments are presumed to have been validated for size and order for the np.uint16 type. Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution. high : int Highest (signed) integer to be drawn from the distribution. size : int or tuple of ints Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. rngstate : encapsulated pointer to rk_state The specific type depends on the python version. In Python 2 it is a PyCObject, in Python 3 a PyCapsule object. Returns ------- out : python integer or ndarray of np.uint16 `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. _rand_uint8(low, high, size, rngstate) Return random np.uint8 integers between ``low`` and ``high``, inclusive. Return random integers from the "discrete uniform" distribution in the closed interval [``low``, ``high``). On entry the arguments are presumed to have been validated for size and order for the np.uint8 type. Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution. high : int Highest (signed) integer to be drawn from the distribution. size : int or tuple of ints Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. rngstate : encapsulated pointer to rk_state The specific type depends on the python version. In Python 2 it is a PyCObject, in Python 3 a PyCapsule object. Returns ------- out : python integer or ndarray of np.uint8 `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. _rand_int64(low, high, size, rngstate) Return random np.int64 integers between ``low`` and ``high``, inclusive. Return random integers from the "discrete uniform" distribution in the closed interval [``low``, ``high``). On entry the arguments are presumed to have been validated for size and order for the np.int64 type. Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution. high : int Highest (signed) integer to be drawn from the distribution. size : int or tuple of ints Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. rngstate : encapsulated pointer to rk_state The specific type depends on the python version. In Python 2 it is a PyCObject, in Python 3 a PyCapsule object. Returns ------- out : python integer or ndarray of np.int64 `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. _rand_int32(low, high, size, rngstate) Return random np.int32 integers between ``low`` and ``high``, inclusive. Return random integers from the "discrete uniform" distribution in the closed interval [``low``, ``high``). On entry the arguments are presumed to have been validated for size and order for the np.int32 type. Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution. high : int Highest (signed) integer to be drawn from the distribution. size : int or tuple of ints Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. rngstate : encapsulated pointer to rk_state The specific type depends on the python version. In Python 2 it is a PyCObject, in Python 3 a PyCapsule object. Returns ------- out : python integer or ndarray of np.int32 `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. _rand_int16(low, high, size, rngstate) Return random np.int16 integers between ``low`` and ``high``, inclusive. Return random integers from the "discrete uniform" distribution in the closed interval [``low``, ``high``). On entry the arguments are presumed to have been validated for size and order for the np.int16 type. Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution. high : int Highest (signed) integer to be drawn from the distribution. size : int or tuple of ints Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. rngstate : encapsulated pointer to rk_state The specific type depends on the python version. In Python 2 it is a PyCObject, in Python 3 a PyCapsule object. Returns ------- out : python integer or ndarray of np.int16 `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. _rand_int8(low, high, size, rngstate) Return random np.int8 integers between ``low`` and ``high``, inclusive. Return random integers from the "discrete uniform" distribution in the closed interval [``low``, ``high``). On entry the arguments are presumed to have been validated for size and order for the np.int8 type. Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution. high : int Highest (signed) integer to be drawn from the distribution. size : int or tuple of ints Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. rngstate : encapsulated pointer to rk_state The specific type depends on the python version. In Python 2 it is a PyCObject, in Python 3 a PyCapsule object. Returns ------- out : python integer or ndarray of np.int8 `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. _rand_bool(low, high, size, rngstate) Return random np.bool_ integers between ``low`` and ``high``, inclusive. Return random integers from the "discrete uniform" distribution in the closed interval [``low``, ``high``). On entry the arguments are presumed to have been validated for size and order for the np.bool_ type. Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution. high : int Highest (signed) integer to be drawn from the distribution. size : int or tuple of ints Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. rngstate : encapsulated pointer to rk_state The specific type depends on the python version. In Python 2 it is a PyCObject, in Python 3 a PyCapsule object. Returns ------- out : python integer or ndarray of np.bool_ `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. multivariate_normal__RandomState_ctorDeprecationWarningnegative_binomial_shape_from_sizestandard_normalstandard_cauchyrandom_integerspoisson_lam_maxdummy_threadingstandard_gammahypergeometricRuntimeWarningrandom_samplegreater_equalcount_nonzero_randint_typeOverflowErrorsearchsortedreturn_indexnoncentral_f_rand_uint64_rand_uint32_rand_uint16permutationmultinomialexponentialcheck_valid_rand_uint8_rand_int64_rand_int32_rand_int16ImportErrortriangularstandard_tlogical_orless_equalissubdtypeempty_likearray_data_rand_int8_rand_bool__import__ValueErrorthreadingset_statelogserieslognormalget_stategeometricdirichletchisquarebroadcast__enter__TypeErrorExceptionwarningsvonmisessubtractrngstatereversedrayleighoperatorlogisticitemsizeisfinitefloatingbinomialallclose__test____main____exit__weibulluniformstridessignbitshufflereshapereplacerandintpoissonnsamplendarraylaplaceintegergreaterfloat64castingasarrayMT19937unsafeuniqueuint64uint32uint16reducerandomparetonormalmtrandignoregumbelformatcumsumctypeschoiceastypearangezerosuint8statesigmashapescalerightravelrangerandnraisepvalspowernumpyngoodkappaisnanint64int32int16indexiinfogammafinfoequalemptydtypedfnumdfdenbytesbool_arrayalpha_randzipfwarnwalduinttakesqrtsortsizesideseedsafertolrandprodnoncndimnbadnamemodemeanlonglessleftitemintpint8highdatacopyboolbetaatolLocktolsvdrngoutoffmaxlowloclamintepsdotcovcntbufanyalladdnpmudf permutation(x) Randomly permute a sequence, or return a permuted range. If `x` is a multi-dimensional array, it is only shuffled along its first index. Parameters ---------- x : int or array_like If `x` is an integer, randomly permute ``np.arange(x)``. If `x` is an array, make a copy and shuffle the elements randomly. Returns ------- out : ndarray Permuted sequence or array range. Examples -------- >>> np.random.permutation(10) array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) >>> np.random.permutation([1, 4, 9, 12, 15]) array([15, 1, 9, 4, 12]) >>> arr = np.arange(9).reshape((3, 3)) >>> np.random.permutation(arr) array([[6, 7, 8], [0, 1, 2], [3, 4, 5]]) RandomState.permutation (line 4847) shuffle(x) Modify a sequence in-place by shuffling its contents. This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same. Parameters ---------- x : array_like The array or list to be shuffled. Returns ------- None Examples -------- >>> arr = np.arange(10) >>> np.random.shuffle(arr) >>> arr [1 7 5 2 9 4 3 6 0 8] Multi-dimensional arrays are only shuffled along the first axis: >>> arr = np.arange(9).reshape((3, 3)) >>> np.random.shuffle(arr) >>> arr array([[3, 4, 5], [6, 7, 8], [0, 1, 2]]) RandomState.shuffle (line 4759) dirichlet(alpha, size=None) Draw samples from the Dirichlet distribution. Draw `size` samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. Dirichlet pdf is the conjugate prior of a multinomial in Bayesian inference. Parameters ---------- alpha : array Parameter of the distribution (k dimension for sample of dimension k). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- samples : ndarray, The drawn samples, of shape (size, alpha.ndim). Notes ----- .. math:: X \approx \prod_{i=1}^{k}{x^{\alpha_i-1}_i} Uses the following property for computation: for each dimension, draw a random sample y_i from a standard gamma generator of shape `alpha_i`, then :math:`X = \frac{1}{\sum_{i=1}^k{y_i}} (y_1, \ldots, y_n)` is Dirichlet distributed. References ---------- .. [1] David McKay, "Information Theory, Inference and Learning Algorithms," chapter 23, http://www.inference.phy.cam.ac.uk/mackay/ .. [2] Wikipedia, "Dirichlet distribution", http://en.wikipedia.org/wiki/Dirichlet_distribution Examples -------- Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces. >>> s = np.random.dirichlet((10, 5, 3), 20).transpose() >>> plt.barh(range(20), s[0]) >>> plt.barh(range(20), s[1], left=s[0], color='g') >>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r') >>> plt.title("Lengths of Strings") RandomState.dirichlet (line 4643) multinomial(n, pvals, size=None) Draw samples from a multinomial distribution. The multinomial distribution is a multivariate generalisation of the binomial distribution. Take an experiment with one of ``p`` possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents `n` such experiments. Its values, ``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the outcome was ``i``. Parameters ---------- n : int Number of experiments. pvals : sequence of floats, length p Probabilities of each of the ``p`` different outcomes. These should sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as ``sum(pvals[:-1]) <= 1)``. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : ndarray The drawn samples, of shape *size*, if that was provided. If not, the shape is ``(N,)``. In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. Examples -------- Throw a dice 20 times: >>> np.random.multinomial(20, [1/6.]*6, size=1) array([[4, 1, 7, 5, 2, 1]]) It landed 4 times on 1, once on 2, etc. Now, throw the dice 20 times, and 20 times again: >>> np.random.multinomial(20, [1/6.]*6, size=2) array([[3, 4, 3, 3, 4, 3], [2, 4, 3, 4, 0, 7]]) For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc. A loaded die is more likely to land on number 6: >>> np.random.multinomial(100, [1/7.]*5 + [2/7.]) array([11, 16, 14, 17, 16, 26]) The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so: >>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT array([38, 62]) not like: >>> np.random.multinomial(100, [1.0, 2.0]) # WRONG array([100, 0]) RandomState.multinomial (line 4530) multivariate_normal(mean, cov[, size, check_valid, tol]) Draw random samples from a multivariate normal distribution. The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or "center") and variance (standard deviation, or "width," squared) of the one-dimensional normal distribution. Parameters ---------- mean : 1-D array_like, of length N Mean of the N-dimensional distribution. cov : 2-D array_like, of shape (N, N) Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling. size : int or tuple of ints, optional Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are generated, and packed in an `m`-by-`n`-by-`k` arrangement. Because each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``. If no shape is specified, a single (`N`-D) sample is returned. check_valid : { 'warn', 'raise', 'ignore' }, optional Behavior when the covariance matrix is not positive semidefinite. tol : float, optional Tolerance when checking the singular values in covariance matrix. Returns ------- out : ndarray The drawn samples, of shape *size*, if that was provided. If not, the shape is ``(N,)``. In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. Notes ----- The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution. Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, :math:`X = [x_1, x_2, ... x_N]`. The covariance matrix element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`. The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its "spread"). Instead of specifying the full covariance matrix, popular approximations include: - Spherical covariance (`cov` is a multiple of the identity matrix) - Diagonal covariance (`cov` has non-negative elements, and only on the diagonal) This geometrical property can be seen in two dimensions by plotting generated data-points: >>> mean = [0, 0] >>> cov = [[1, 0], [0, 100]] # diagonal covariance Diagonal covariance means that points are oriented along x or y-axis: >>> import matplotlib.pyplot as plt >>> x, y = np.random.multivariate_normal(mean, cov, 5000).T >>> plt.plot(x, y, 'x') >>> plt.axis('equal') >>> plt.show() Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed. References ---------- .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic Processes," 3rd ed., New York: McGraw-Hill, 1991. .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern Classification," 2nd ed., New York: Wiley, 2001. Examples -------- >>> mean = (1, 2) >>> cov = [[1, 0], [0, 1]] >>> x = np.random.multivariate_normal(mean, cov, (3, 3)) >>> x.shape (3, 3, 2) The following is probably true, given that 0.6 is roughly twice the standard deviation: >>> list((x[0,0,:] - mean) < 0.6) [True, True] RandomState.multivariate_normal (line 4369) logseries(p, size=None) Draw samples from a logarithmic series distribution. Samples are drawn from a log series distribution with specified shape parameter, 0 < ``p`` < 1. Parameters ---------- p : float or array_like of floats Shape parameter for the distribution. Must be in the range (0, 1). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logarithmic series distribution. See Also -------- scipy.stats.logser : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Log Series distribution is .. math:: P(k) = \frac{-p^k}{k \ln(1-p)}, where p = probability. The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3]. References ---------- .. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9). .. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58. .. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994. .. [4] Wikipedia, "Logarithmic distribution", http://en.wikipedia.org/wiki/Logarithmic_distribution Examples -------- Draw samples from the distribution: >>> a = .6 >>> s = np.random.logseries(a, 10000) >>> count, bins, ignored = plt.hist(s) # plot against distribution >>> def logseries(k, p): ... return -p**k/(k*log(1-p)) >>> plt.plot(bins, logseries(bins, a)*count.max()/ logseries(bins, a).max(), 'r') >>> plt.show() RandomState.logseries (line 4272) hypergeometric(ngood, nbad, nsample, size=None) Draw samples from a Hypergeometric distribution. Samples are drawn from a hypergeometric distribution with specified parameters, ngood (ways to make a good selection), nbad (ways to make a bad selection), and nsample = number of items sampled, which is less than or equal to the sum ngood + nbad. Parameters ---------- ngood : int or array_like of ints Number of ways to make a good selection. Must be nonnegative. nbad : int or array_like of ints Number of ways to make a bad selection. Must be nonnegative. nsample : int or array_like of ints Number of items sampled. Must be at least 1 and at most ``ngood + nbad``. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``ngood``, ``nbad``, and ``nsample`` are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized hypergeometric distribution. See Also -------- scipy.stats.hypergeom : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Hypergeometric distribution is .. math:: P(x) = \frac{\binom{m}{n}\binom{N-m}{n-x}}{\binom{N}{n}}, where :math:`0 \le x \le m` and :math:`n+m-N \le x \le n` for P(x) the probability of x successes, n = ngood, m = nbad, and N = number of samples. Consider an urn with black and white marbles in it, ngood of them black and nbad are white. If you draw nsample balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample. Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial. References ---------- .. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972. .. [2] Weisstein, Eric W. "Hypergeometric Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html .. [3] Wikipedia, "Hypergeometric distribution", http://en.wikipedia.org/wiki/Hypergeometric_distribution Examples -------- Draw samples from the distribution: >>> ngood, nbad, nsamp = 100, 2, 10 # number of good, number of bad, and number of samples >>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000) >>> hist(s) # note that it is very unlikely to grab both bad items Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color? >>> s = np.random.hypergeometric(15, 15, 15, 100000) >>> sum(s>=12)/100000. + sum(s<=3)/100000. # answer = 0.003 ... pretty unlikely! RandomState.hypergeometric (line 4150) geometric(p, size=None) Draw samples from the geometric distribution. Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, ``k = 1, 2, ...``. The probability mass function of the geometric distribution is .. math:: f(k) = (1 - p)^{k - 1} p where `p` is the probability of success of an individual trial. Parameters ---------- p : float or array_like of floats The probability of success of an individual trial. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized geometric distribution. Examples -------- Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35: >>> z = np.random.geometric(p=0.35, size=10000) How many trials succeeded after a single run? >>> (z == 1).sum() / 10000. 0.34889999999999999 #random RandomState.geometric (line 4082) zipf(a, size=None) Draw samples from a Zipf distribution. Samples are drawn from a Zipf distribution with specified parameter `a` > 1. The Zipf distribution (also known as the zeta distribution) is a continuous probability distribution that satisfies Zipf's law: the frequency of an item is inversely proportional to its rank in a frequency table. Parameters ---------- a : float or array_like of floats Distribution parameter. Should be greater than 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Zipf distribution. See Also -------- scipy.stats.zipf : probability density function, distribution, or cumulative density function, etc. Notes ----- The probability density for the Zipf distribution is .. math:: p(x) = \frac{x^{-a}}{\zeta(a)}, where :math:`\zeta` is the Riemann Zeta function. It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table. References ---------- .. [1] Zipf, G. K., "Selected Studies of the Principle of Relative Frequency in Language," Cambridge, MA: Harvard Univ. Press, 1932. Examples -------- Draw samples from the distribution: >>> a = 2. # parameter >>> s = np.random.zipf(a, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy import special Truncate s values at 50 so plot is interesting: >>> count, bins, ignored = plt.hist(s[s<50], 50, normed=True) >>> x = np.arange(1., 50.) >>> y = x**(-a) / special.zetac(a) >>> plt.plot(x, y/max(y), linewidth=2, color='r') >>> plt.show() RandomState.zipf (line 3991) poisson(lam=1.0, size=None) Draw samples from a Poisson distribution. The Poisson distribution is the limit of the binomial distribution for large N. Parameters ---------- lam : float or array_like of floats Expectation of interval, should be >= 0. A sequence of expectation intervals must be broadcastable over the requested size. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``lam`` is a scalar. Otherwise, ``np.array(lam).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Poisson distribution. Notes ----- The Poisson distribution .. math:: f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!} For events with an expected separation :math:`\lambda` the Poisson distribution :math:`f(k; \lambda)` describes the probability of :math:`k` events occurring within the observed interval :math:`\lambda`. Because the output is limited to the range of the C long type, a ValueError is raised when `lam` is within 10 sigma of the maximum representable value. References ---------- .. [1] Weisstein, Eric W. "Poisson Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html .. [2] Wikipedia, "Poisson distribution", http://en.wikipedia.org/wiki/Poisson_distribution Examples -------- Draw samples from the distribution: >>> import numpy as np >>> s = np.random.poisson(5, 10000) Display histogram of the sample: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 14, normed=True) >>> plt.show() Draw each 100 values for lambda 100 and 500: >>> s = np.random.poisson(lam=(100., 500.), size=(100, 2)) RandomState.poisson (line 3903) negative_binomial(n, p, size=None) Draw samples from a negative binomial distribution. Samples are drawn from a negative binomial distribution with specified parameters, `n` trials and `p` probability of success where `n` is an integer > 0 and `p` is in the interval [0, 1]. Parameters ---------- n : int or array_like of ints Parameter of the distribution, > 0. Floats are also accepted, but they will be truncated to integers. p : float or array_like of floats Parameter of the distribution, >= 0 and <=1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``n`` and ``p`` are both scalars. Otherwise, ``np.broadcast(n, p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of trials it took to achieve n - 1 successes, N - (n - 1) failures, and a success on the, (N + n)th trial. Notes ----- The probability density for the negative binomial distribution is .. math:: P(N;n,p) = \binom{N+n-1}{n-1}p^{n}(1-p)^{N}, where :math:`n-1` is the number of successes, :math:`p` is the probability of success, and :math:`N+n-1` is the number of trials. The negative binomial distribution gives the probability of n-1 successes and N failures in N+n-1 trials, and success on the (N+n)th trial. If one throws a die repeatedly until the third time a "1" appears, then the probability distribution of the number of non-"1"s that appear before the third "1" is a negative binomial distribution. References ---------- .. [1] Weisstein, Eric W. "Negative Binomial Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html .. [2] Wikipedia, "Negative binomial distribution", http://en.wikipedia.org/wiki/Negative_binomial_distribution Examples -------- Draw samples from the distribution: A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.? >>> s = np.random.negative_binomial(1, 0.1, 100000) >>> for i in range(1, 11): ... probability = sum(s>> n, p = 10, .5 # number of trials, probability of each trial >>> s = np.random.binomial(n, p, 1000) # result of flipping a coin 10 times, tested 1000 times. A real world example. A company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of 0.1. All nine wells fail. What is the probability of that happening? Let's do 20,000 trials of the model, and count the number that generate zero positive results. >>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000. # answer = 0.38885, or 38%. RandomState.binomial (line 3686) triangular(left, mode, right, size=None) Draw samples from the triangular distribution over the interval ``[left, right]``. The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf. Parameters ---------- left : float or array_like of floats Lower limit. mode : float or array_like of floats The value where the peak of the distribution occurs. The value should fulfill the condition ``left <= mode <= right``. right : float or array_like of floats Upper limit, should be larger than `left`. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``left``, ``mode``, and ``right`` are all scalars. Otherwise, ``np.broadcast(left, mode, right).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized triangular distribution. Notes ----- The probability density function for the triangular distribution is .. math:: P(x;l, m, r) = \begin{cases} \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases} The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations. References ---------- .. [1] Wikipedia, "Triangular distribution" http://en.wikipedia.org/wiki/Triangular_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200, ... normed=True) >>> plt.show() RandomState.triangular (line 3592) wald(mean, scale, size=None) Draw samples from a Wald, or inverse Gaussian, distribution. As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal. The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time. Parameters ---------- mean : float or array_like of floats Distribution mean, should be > 0. scale : float or array_like of floats Scale parameter, should be >= 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(mean, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Wald distribution. Notes ----- The probability density function for the Wald distribution is .. math:: P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x} As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes. References ---------- .. [1] Brighton Webs Ltd., Wald Distribution, http://www.brighton-webs.co.uk/distributions/wald.asp .. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian Distribution: Theory : Methodology, and Applications", CRC Press, 1988. .. [3] Wikipedia, "Wald distribution" http://en.wikipedia.org/wiki/Wald_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, normed=True) >>> plt.show() RandomState.wald (line 3505) rayleigh(scale=1.0, size=None) Draw samples from a Rayleigh distribution. The :math:`\chi` and Weibull distributions are generalizations of the Rayleigh. Parameters ---------- scale : float or array_like of floats, optional Scale, also equals the mode. Should be >= 0. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Rayleigh distribution. Notes ----- The probability density function for the Rayleigh distribution is .. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}} The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution. References ---------- .. [1] Brighton Webs Ltd., "Rayleigh Distribution," http://www.brighton-webs.co.uk/distributions/rayleigh.asp .. [2] Wikipedia, "Rayleigh distribution" http://en.wikipedia.org/wiki/Rayleigh_distribution Examples -------- Draw values from the distribution and plot the histogram >>> values = hist(np.random.rayleigh(3, 100000), bins=200, normed=True) Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters? >>> meanvalue = 1 >>> modevalue = np.sqrt(2 / np.pi) * meanvalue >>> s = np.random.rayleigh(modevalue, 1000000) The percentage of waves larger than 3 meters is: >>> 100.*sum(s>3)/1000000. 0.087300000000000003 RandomState.rayleigh (line 3426) lognormal(mean=0.0, sigma=1.0, size=None) Draw samples from a log-normal distribution. Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from. Parameters ---------- mean : float or array_like of floats, optional Mean value of the underlying normal distribution. Default is 0. sigma : float or array_like of floats, optional Standard deviation of the underlying normal distribution. Should be greater than zero. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``sigma`` are both scalars. Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized log-normal distribution. See Also -------- scipy.stats.lognorm : probability density function, distribution, cumulative density function, etc. Notes ----- A variable `x` has a log-normal distribution if `log(x)` is normally distributed. The probability density function for the log-normal distribution is: .. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})} where :math:`\mu` is the mean and :math:`\sigma` is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the *product* of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the *sum* of a large number of independent, identically-distributed variables. References ---------- .. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal Distributions across the Sciences: Keys and Clues," BioScience, Vol. 51, No. 5, May, 2001. http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf .. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme Values," Basel: Birkhauser Verlag, 2001, pp. 31-32. Examples -------- Draw samples from the distribution: >>> mu, sigma = 3., 1. # mean and standard deviation >>> s = np.random.lognormal(mu, sigma, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 100, normed=True, align='mid') >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, linewidth=2, color='r') >>> plt.axis('tight') >>> plt.show() Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function. >>> # Generate a thousand samples: each is the product of 100 random >>> # values, drawn from a normal distribution. >>> b = [] >>> for i in range(1000): ... a = 10. + np.random.random(100) ... b.append(np.product(a)) >>> b = np.array(b) / np.min(b) # scale values to be positive >>> count, bins, ignored = plt.hist(b, 100, normed=True, align='mid') >>> sigma = np.std(np.log(b)) >>> mu = np.mean(np.log(b)) >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, color='r', linewidth=2) >>> plt.show() RandomState.lognormal (line 3302) logistic(loc=0.0, scale=1.0, size=None) Draw samples from a logistic distribution. Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0). Parameters ---------- loc : float or array_like of floats, optional Parameter of the distribution. Default is 0. scale : float or array_like of floats, optional Parameter of the distribution. Should be greater than zero. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logistic distribution. See Also -------- scipy.stats.logistic : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Logistic distribution is .. math:: P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2}, where :math:`\mu` = location and :math:`s` = scale. The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable. References ---------- .. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields," Birkhauser Verlag, Basel, pp 132-133. .. [2] Weisstein, Eric W. "Logistic Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html .. [3] Wikipedia, "Logistic-distribution", http://en.wikipedia.org/wiki/Logistic_distribution Examples -------- Draw samples from the distribution: >>> loc, scale = 10, 1 >>> s = np.random.logistic(loc, scale, 10000) >>> count, bins, ignored = plt.hist(s, bins=50) # plot against distribution >>> def logist(x, loc, scale): ... return exp((loc-x)/scale)/(scale*(1+exp((loc-x)/scale))**2) >>> plt.plot(bins, logist(bins, loc, scale)*count.max()/\ ... logist(bins, loc, scale).max()) >>> plt.show() RandomState.logistic (line 3209) gumbel(loc=0.0, scale=1.0, size=None) Draw samples from a Gumbel distribution. Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below. Parameters ---------- loc : float or array_like of floats, optional The location of the mode of the distribution. Default is 0. scale : float or array_like of floats, optional The scale parameter of the distribution. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Gumbel distribution. See Also -------- scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull Notes ----- The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with "exponential-like" tails. The probability density for the Gumbel distribution is .. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}}, where :math:`\mu` is the mode, a location parameter, and :math:`\beta` is the scale parameter. The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a "fat-tailed" distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events. It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet. The function has a mean of :math:`\mu + 0.57721\beta` and a variance of :math:`\frac{\pi^2}{6}\beta^2`. References ---------- .. [1] Gumbel, E. J., "Statistics of Extremes," New York: Columbia University Press, 1958. .. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields," Basel: Birkhauser Verlag, 2001. Examples -------- Draw samples from the distribution: >>> mu, beta = 0, 0.1 # location and scale >>> s = np.random.gumbel(mu, beta, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, normed=True) >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp( -np.exp( -(bins - mu) /beta) ), ... linewidth=2, color='r') >>> plt.show() Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian: >>> means = [] >>> maxima = [] >>> for i in range(0,1000) : ... a = np.random.normal(mu, beta, 1000) ... means.append(a.mean()) ... maxima.append(a.max()) >>> count, bins, ignored = plt.hist(maxima, 30, normed=True) >>> beta = np.std(maxima) * np.sqrt(6) / np.pi >>> mu = np.mean(maxima) - 0.57721*beta >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp(-np.exp(-(bins - mu)/beta)), ... linewidth=2, color='r') >>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi)) ... * np.exp(-(bins - mu)**2 / (2 * beta**2)), ... linewidth=2, color='g') >>> plt.show() RandomState.gumbel (line 3078) laplace(loc=0.0, scale=1.0, size=None) Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay). The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables. Parameters ---------- loc : float or array_like of floats, optional The position, :math:`\mu`, of the distribution peak. Default is 0. scale : float or array_like of floats, optional :math:`\lambda`, the exponential decay. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Laplace distribution. Notes ----- It has the probability density function .. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right). The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] Kotz, Samuel, et. al. "The Laplace Distribution and Generalizations, " Birkhauser, 2001. .. [3] Weisstein, Eric W. "Laplace Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html .. [4] Wikipedia, "Laplace distribution", http://en.wikipedia.org/wiki/Laplace_distribution Examples -------- Draw samples from the distribution >>> loc, scale = 0., 1. >>> s = np.random.laplace(loc, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, normed=True) >>> x = np.arange(-8., 8., .01) >>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale) >>> plt.plot(x, pdf) Plot Gaussian for comparison: >>> g = (1/(scale * np.sqrt(2 * np.pi)) * ... np.exp(-(x - loc)**2 / (2 * scale**2))) >>> plt.plot(x,g) RandomState.laplace (line 2980) power(a, size=None) Draws samples in [0, 1] from a power distribution with positive exponent a - 1. Also known as the power function distribution. Parameters ---------- a : float or array_like of floats Parameter of the distribution. Should be greater than zero. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized power distribution. Raises ------ ValueError If a < 1. Notes ----- The probability density function is .. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0. The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution. It is used, for example, in modeling the over-reporting of insurance claims. References ---------- .. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions in economics and actuarial sciences", Wiley, 2003. .. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions", National Institute of Standards and Technology Handbook Series, June 2003. http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf Examples -------- Draw samples from the distribution: >>> a = 5. # shape >>> samples = 1000 >>> s = np.random.power(a, samples) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, bins=30) >>> x = np.linspace(0, 1, 100) >>> y = a*x**(a-1.) >>> normed_y = samples*np.diff(bins)[0]*y >>> plt.plot(x, normed_y) >>> plt.show() Compare the power function distribution to the inverse of the Pareto. >>> from scipy import stats >>> rvs = np.random.power(5, 1000000) >>> rvsp = np.random.pareto(5, 1000000) >>> xx = np.linspace(0,1,100) >>> powpdf = stats.powerlaw.pdf(xx,5) >>> plt.figure() >>> plt.hist(rvs, bins=50, normed=True) >>> plt.plot(xx,powpdf,'r-') >>> plt.title('np.random.power(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, normed=True) >>> plt.plot(xx,powpdf,'r-') >>> plt.title('inverse of 1 + np.random.pareto(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, normed=True) >>> plt.plot(xx,powpdf,'r-') >>> plt.title('inverse of stats.pareto(5)') RandomState.power (line 2869) weibull(a, size=None) Draw samples from a Weibull distribution. Draw samples from a 1-parameter Weibull distribution with the given shape parameter `a`. .. math:: X = (-ln(U))^{1/a} Here, U is drawn from the uniform distribution over (0,1]. The more common 2-parameter Weibull, including a scale parameter :math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`. Parameters ---------- a : float or array_like of floats Shape of the distribution. Should be greater than zero. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Weibull distribution. See Also -------- scipy.stats.weibull_max scipy.stats.weibull_min scipy.stats.genextreme gumbel Notes ----- The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions. The probability density for the Weibull distribution is .. math:: p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a}, where :math:`a` is the shape and :math:`\lambda` the scale. The function has its peak (the mode) at :math:`\lambda(\frac{a-1}{a})^{1/a}`. When ``a = 1``, the Weibull distribution reduces to the exponential distribution. References ---------- .. [1] Waloddi Weibull, Royal Technical University, Stockholm, 1939 "A Statistical Theory Of The Strength Of Materials", Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm. .. [2] Waloddi Weibull, "A Statistical Distribution Function of Wide Applicability", Journal Of Applied Mechanics ASME Paper 1951. .. [3] Wikipedia, "Weibull distribution", http://en.wikipedia.org/wiki/Weibull_distribution Examples -------- Draw samples from the distribution: >>> a = 5. # shape >>> s = np.random.weibull(a, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> x = np.arange(1,100.)/50. >>> def weib(x,n,a): ... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a) >>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000)) >>> x = np.arange(1,100.)/50. >>> scale = count.max()/weib(x, 1., 5.).max() >>> plt.plot(x, weib(x, 1., 5.)*scale) >>> plt.show() RandomState.weibull (line 2759) pareto(a, size=None) Draw samples from a Pareto II or Lomax distribution with specified shape. The Lomax or Pareto II distribution is a shifted Pareto distribution. The classical Pareto distribution can be obtained from the Lomax distribution by adding 1 and multiplying by the scale parameter ``m`` (see Notes). The smallest value of the Lomax distribution is zero while for the classical Pareto distribution it is ``mu``, where the standard Pareto distribution has location ``mu = 1``. Lomax can also be considered as a simplified version of the Generalized Pareto distribution (available in SciPy), with the scale set to one and the location set to zero. The Pareto distribution must be greater than zero, and is unbounded above. It is also known as the "80-20 rule". In this distribution, 80 percent of the weights are in the lowest 20 percent of the range, while the other 20 percent fill the remaining 80 percent of the range. Parameters ---------- a : float or array_like of floats Shape of the distribution. Should be greater than zero. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Pareto distribution. See Also -------- scipy.stats.lomax : probability density function, distribution or cumulative density function, etc. scipy.stats.genpareto : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Pareto distribution is .. math:: p(x) = \frac{am^a}{x^{a+1}} where :math:`a` is the shape and :math:`m` the scale. The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution useful in many real world problems. Outside the field of economics it is generally referred to as the Bradford distribution. Pareto developed the distribution to describe the distribution of wealth in an economy. It has also found use in insurance, web page access statistics, oil field sizes, and many other problems, including the download frequency for projects in Sourceforge [1]_. It is one of the so-called "fat-tailed" distributions. References ---------- .. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects. .. [2] Pareto, V. (1896). Course of Political Economy. Lausanne. .. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30. .. [4] Wikipedia, "Pareto distribution", http://en.wikipedia.org/wiki/Pareto_distribution Examples -------- Draw samples from the distribution: >>> a, m = 3., 2. # shape and mode >>> s = (np.random.pareto(a, 1000) + 1) * m Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 100, normed=True) >>> fit = a*m**a / bins**(a+1) >>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r') >>> plt.show() RandomState.pareto (line 2649) vonmises(mu, kappa, size=None) Draw samples from a von Mises distribution. Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi]. The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution. Parameters ---------- mu : float or array_like of floats Mode ("center") of the distribution. kappa : float or array_like of floats Dispersion of the distribution, has to be >=0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mu`` and ``kappa`` are both scalars. Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized von Mises distribution. See Also -------- scipy.stats.vonmises : probability density function, distribution, or cumulative density function, etc. Notes ----- The probability density for the von Mises distribution is .. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}, where :math:`\mu` is the mode and :math:`\kappa` the dispersion, and :math:`I_0(\kappa)` is the modified Bessel function of order 0. The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] von Mises, R., "Mathematical Theory of Probability and Statistics", New York: Academic Press, 1964. Examples -------- Draw samples from the distribution: >>> mu, kappa = 0.0, 4.0 # mean and dispersion >>> s = np.random.vonmises(mu, kappa, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import i0 >>> plt.hist(s, 50, normed=True) >>> x = np.linspace(-np.pi, np.pi, num=51) >>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa)) >>> plt.plot(x, y, linewidth=2, color='r') >>> plt.show() RandomState.vonmises (line 2551) standard_t(df, size=None) Draw samples from a standard Student's t distribution with `df` degrees of freedom. A special case of the hyperbolic distribution. As `df` gets large, the result resembles that of the standard normal distribution (`standard_normal`). Parameters ---------- df : int or array_like of ints Degrees of freedom, should be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard Student's t distribution. Notes ----- The probability density function for the t distribution is .. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2} The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean. The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student. References ---------- .. [1] Dalgaard, Peter, "Introductory Statistics With R", Springer, 2002. .. [2] Wikipedia, "Student's t-distribution" http://en.wikipedia.org/wiki/Student's_t-distribution Examples -------- From Dalgaard page 83 [1]_, suppose the daily energy intake for 11 women in Kj is: >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \ ... 7515, 8230, 8770]) Does their energy intake deviate systematically from the recommended value of 7725 kJ? We have 10 degrees of freedom, so is the sample mean within 95% of the recommended value? >>> s = np.random.standard_t(10, size=100000) >>> np.mean(intake) 6753.636363636364 >>> intake.std(ddof=1) 1142.1232221373727 Calculate the t statistic, setting the ddof parameter to the unbiased value so the divisor in the standard deviation will be degrees of freedom, N-1. >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake))) >>> import matplotlib.pyplot as plt >>> h = plt.hist(s, bins=100, normed=True) For a one-sided t-test, how far out in the distribution does the t statistic appear? >>> np.sum(s>> s = np.random.standard_cauchy(1000000) >>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well >>> plt.hist(s, bins=100) >>> plt.show() RandomState.standard_cauchy (line 2381) noncentral_chisquare(df, nonc, size=None) Draw samples from a noncentral chi-square distribution. The noncentral :math:`\chi^2` distribution is a generalisation of the :math:`\chi^2` distribution. Parameters ---------- df : int or array_like of ints Degrees of freedom, should be > 0 as of NumPy 1.10.0, should be > 1 for earlier versions. nonc : float or array_like of floats Non-centrality, should be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` and ``nonc`` are both scalars. Otherwise, ``np.broadcast(df, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral chi-square distribution. Notes ----- The probability density function for the noncentral Chi-square distribution is .. math:: P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} \P_{Y_{df+2i}}(x), where :math:`Y_{q}` is the Chi-square with q degrees of freedom. In Delhi (2007), it is noted that the noncentral chi-square is useful in bombing and coverage problems, the probability of killing the point target given by the noncentral chi-squared distribution. References ---------- .. [1] Delhi, M.S. Holla, "On a noncentral chi-square distribution in the analysis of weapon systems effectiveness", Metrika, Volume 15, Number 1 / December, 1970. .. [2] Wikipedia, "Noncentral chi-square distribution" http://en.wikipedia.org/wiki/Noncentral_chi-square_distribution Examples -------- Draw values from the distribution and plot the histogram >>> import matplotlib.pyplot as plt >>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000), ... bins=200, normed=True) >>> plt.show() Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare. >>> plt.figure() >>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000), ... bins=np.arange(0., 25, .1), normed=True) >>> values2 = plt.hist(np.random.chisquare(3, 100000), ... bins=np.arange(0., 25, .1), normed=True) >>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob') >>> plt.show() Demonstrate how large values of non-centrality lead to a more symmetric distribution. >>> plt.figure() >>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000), ... bins=200, normed=True) >>> plt.show() RandomState.noncentral_chisquare (line 2277) chisquare(df, size=None) Draw samples from a chi-square distribution. When `df` independent random variables, each with standard normal distributions (mean 0, variance 1), are squared and summed, the resulting distribution is chi-square (see Notes). This distribution is often used in hypothesis testing. Parameters ---------- df : int or array_like of ints Number of degrees of freedom. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized chi-square distribution. Raises ------ ValueError When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``) is given. Notes ----- The variable obtained by summing the squares of `df` independent, standard normally distributed random variables: .. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i is chi-square distributed, denoted .. math:: Q \sim \chi^2_k. The probability density function of the chi-squared distribution is .. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, where :math:`\Gamma` is the gamma function, .. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt. References ---------- .. [1] NIST "Engineering Statistics Handbook" http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Examples -------- >>> np.random.chisquare(2,4) array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) RandomState.chisquare (line 2196) noncentral_f(dfnum, dfden, nonc, size=None) Draw samples from the noncentral F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters > 1. `nonc` is the non-centrality parameter. Parameters ---------- dfnum : int or array_like of ints Parameter, should be > 1. dfden : int or array_like of ints Parameter, should be > 1. nonc : float or array_like of floats Parameter, should be >= 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum``, ``dfden``, and ``nonc`` are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral Fisher distribution. Notes ----- When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic. References ---------- .. [1] Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html .. [2] Wikipedia, "Noncentral F-distribution", http://en.wikipedia.org/wiki/Noncentral_F-distribution Examples -------- In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We'll plot the two probability distributions for comparison. >>> dfnum = 3 # between group deg of freedom >>> dfden = 20 # within groups degrees of freedom >>> nonc = 3.0 >>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000) >>> NF = np.histogram(nc_vals, bins=50, normed=True) >>> c_vals = np.random.f(dfnum, dfden, 1000000) >>> F = np.histogram(c_vals, bins=50, normed=True) >>> plt.plot(F[1][1:], F[0]) >>> plt.plot(NF[1][1:], NF[0]) >>> plt.show() RandomState.noncentral_f (line 2099) f(dfnum, dfden, size=None) Draw samples from an F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters should be greater than zero. The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates. Parameters ---------- dfnum : int or array_like of ints Degrees of freedom in numerator. Should be greater than zero. dfden : int or array_like of ints Degrees of freedom in denominator. Should be greater than zero. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum`` and ``dfden`` are both scalars. Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Fisher distribution. See Also -------- scipy.stats.f : probability density function, distribution or cumulative density function, etc. Notes ----- The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable `dfnum` is the number of samples minus one, the between-groups degrees of freedom, while `dfden` is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups. References ---------- .. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002. .. [2] Wikipedia, "F-distribution", http://en.wikipedia.org/wiki/F-distribution Examples -------- An example from Glantz[1], pp 47-40: Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children's blood glucose levels? Calculating the F statistic from the data gives a value of 36.01. Draw samples from the distribution: >>> dfnum = 1. # between group degrees of freedom >>> dfden = 48. # within groups degrees of freedom >>> s = np.random.f(dfnum, dfden, 1000) The lower bound for the top 1% of the samples is : >>> sort(s)[-10] 7.61988120985 So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level. RandomState.f (line 1992) gamma(shape, scale=1.0, size=None) Draw samples from a Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, `shape` (sometimes designated "k") and `scale` (sometimes designated "theta"), where both parameters are > 0. Parameters ---------- shape : float or array_like of floats The shape of the gamma distribution. Should be greater than zero. scale : float or array_like of floats, optional The scale of the gamma distribution. Should be greater than zero. Default is equal to 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", http://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2) >>> s = np.random.gamma(shape, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps >>> count, bins, ignored = plt.hist(s, 50, normed=True) >>> y = bins**(shape-1)*(np.exp(-bins/scale) / ... (sps.gamma(shape)*scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') >>> plt.show() RandomState.gamma (line 1896) standard_gamma(shape, size=None) Draw samples from a standard Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated "k") and scale=1. Parameters ---------- shape : float or array_like of floats Parameter, should be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` is a scalar. Otherwise, ``np.array(shape).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", http://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 1. # mean and width >>> s = np.random.standard_gamma(shape, 1000000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps >>> count, bins, ignored = plt.hist(s, 50, normed=True) >>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ \ ... (sps.gamma(shape) * scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') >>> plt.show() RandomState.standard_gamma (line 1810) standard_exponential(size=None) Draw samples from the standard exponential distribution. `standard_exponential` is identical to the exponential distribution with a scale parameter of 1. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray Drawn samples. Examples -------- Output a 3x8000 array: >>> n = np.random.standard_exponential((3, 8000)) RandomState.standard_exponential (line 1779) normal(loc=0.0, scale=1.0, size=None) Draw random samples from a normal (Gaussian) distribution. The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2]_, is often called the bell curve because of its characteristic shape (see the example below). The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2]_. Parameters ---------- loc : float or array_like of floats Mean ("centre") of the distribution. scale : float or array_like of floats Standard deviation (spread or "width") of the distribution. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized normal distribution. See Also -------- scipy.stats.norm : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gaussian distribution is .. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} }, where :math:`\mu` is the mean and :math:`\sigma` the standard deviation. The square of the standard deviation, :math:`\sigma^2`, is called the variance. The function has its peak at the mean, and its "spread" increases with the standard deviation (the function reaches 0.607 times its maximum at :math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that `numpy.random.normal` is more likely to return samples lying close to the mean, rather than those far away. References ---------- .. [1] Wikipedia, "Normal distribution", http://en.wikipedia.org/wiki/Normal_distribution .. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability, Random Variables and Random Signal Principles", 4th ed., 2001, pp. 51, 51, 125. Examples -------- Draw samples from the distribution: >>> mu, sigma = 0, 0.1 # mean and standard deviation >>> s = np.random.normal(mu, sigma, 1000) Verify the mean and the variance: >>> abs(mu - np.mean(s)) < 0.01 True >>> abs(sigma - np.std(s, ddof=1)) < 0.01 True Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, normed=True) >>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) * ... np.exp( - (bins - mu)**2 / (2 * sigma**2) ), ... linewidth=2, color='r') >>> plt.show() RandomState.normal (line 1547) standard_normal(size=None) Draw samples from a standard Normal distribution (mean=0, stdev=1). Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray Drawn samples. Examples -------- >>> s = np.random.standard_normal(8000) >>> s array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, #random -0.38672696, -0.4685006 ]) #random >>> s.shape (8000,) >>> s = np.random.standard_normal(size=(3, 4, 2)) >>> s.shape (3, 4, 2) RandomState.standard_normal (line 1514) random_integers(low, high=None, size=None) Random integers of type np.int between `low` and `high`, inclusive. Return random integers of type np.int from the "discrete uniform" distribution in the closed interval [`low`, `high`]. If `high` is None (the default), then results are from [1, `low`]. The np.int type translates to the C long type used by Python 2 for "short" integers and its precision is platform dependent. This function has been deprecated. Use randint instead. .. deprecated:: 1.11.0 Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution (unless ``high=None``, in which case this parameter is the *highest* such integer). high : int, optional If provided, the largest (signed) integer to be drawn from the distribution (see above for behavior if ``high=None``). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : int or ndarray of ints `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. See Also -------- random.randint : Similar to `random_integers`, only for the half-open interval [`low`, `high`), and 0 is the lowest value if `high` is omitted. Notes ----- To sample from N evenly spaced floating-point numbers between a and b, use:: a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.) Examples -------- >>> np.random.random_integers(5) 4 >>> type(np.random.random_integers(5)) >>> np.random.random_integers(5, size=(3,2)) array([[5, 4], [3, 3], [4, 5]]) Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (*i.e.*, from the set :math:`{0, 5/8, 10/8, 15/8, 20/8}`): >>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4. array([ 0.625, 1.25 , 0.625, 0.625, 2.5 ]) Roll two six sided dice 1000 times and sum the results: >>> d1 = np.random.random_integers(1, 6, 1000) >>> d2 = np.random.random_integers(1, 6, 1000) >>> dsums = d1 + d2 Display results as a histogram: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(dsums, 11, normed=True) >>> plt.show() RandomState.random_integers (line 1417) randn(d0, d1, ..., dn) Return a sample (or samples) from the "standard normal" distribution. If positive, int_like or int-convertible arguments are provided, `randn` generates an array of shape ``(d0, d1, ..., dn)``, filled with random floats sampled from a univariate "normal" (Gaussian) distribution of mean 0 and variance 1 (if any of the :math:`d_i` are floats, they are first converted to integers by truncation). A single float randomly sampled from the distribution is returned if no argument is provided. This is a convenience function. If you want an interface that takes a tuple as the first argument, use `numpy.random.standard_normal` instead. Parameters ---------- d0, d1, ..., dn : int, optional The dimensions of the returned array, should be all positive. If no argument is given a single Python float is returned. Returns ------- Z : ndarray or float A ``(d0, d1, ..., dn)``-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied. See Also -------- random.standard_normal : Similar, but takes a tuple as its argument. Notes ----- For random samples from :math:`N(\mu, \sigma^2)`, use: ``sigma * np.random.randn(...) + mu`` Examples -------- >>> np.random.randn() 2.1923875335537315 #random Two-by-four array of samples from N(3, 6.25): >>> 2.5 * np.random.randn(2, 4) + 3 array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], #random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) #random RandomState.randn (line 1360) rand(d0, d1, ..., dn) Random values in a given shape. Create an array of the given shape and populate it with random samples from a uniform distribution over ``[0, 1)``. Parameters ---------- d0, d1, ..., dn : int, optional The dimensions of the returned array, should all be positive. If no argument is given a single Python float is returned. Returns ------- out : ndarray, shape ``(d0, d1, ..., dn)`` Random values. See Also -------- random Notes ----- This is a convenience function. If you want an interface that takes a shape-tuple as the first argument, refer to np.random.random_sample . Examples -------- >>> np.random.rand(3,2) array([[ 0.14022471, 0.96360618], #random [ 0.37601032, 0.25528411], #random [ 0.49313049, 0.94909878]]) #random RandomState.rand (line 1316) uniform(low=0.0, high=1.0, size=None) Draw samples from a uniform distribution. Samples are uniformly distributed over the half-open interval ``[low, high)`` (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by `uniform`. Parameters ---------- low : float or array_like of floats, optional Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0. high : float or array_like of floats Upper boundary of the output interval. All values generated will be less than high. The default value is 1.0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``low`` and ``high`` are both scalars. Otherwise, ``np.broadcast(low, high).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized uniform distribution. See Also -------- randint : Discrete uniform distribution, yielding integers. random_integers : Discrete uniform distribution over the closed interval ``[low, high]``. random_sample : Floats uniformly distributed over ``[0, 1)``. random : Alias for `random_sample`. rand : Convenience function that accepts dimensions as input, e.g., ``rand(2,2)`` would generate a 2-by-2 array of floats, uniformly distributed over ``[0, 1)``. Notes ----- The probability density function of the uniform distribution is .. math:: p(x) = \frac{1}{b - a} anywhere within the interval ``[a, b)``, and zero elsewhere. When ``high`` == ``low``, values of ``low`` will be returned. If ``high`` < ``low``, the results are officially undefined and may eventually raise an error, i.e. do not rely on this function to behave when passed arguments satisfying that inequality condition. Examples -------- Draw samples from the distribution: >>> s = np.random.uniform(-1,0,1000) All values are within the given interval: >>> np.all(s >= -1) True >>> np.all(s < 0) True Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 15, normed=True) >>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r') >>> plt.show() RandomState.uniform (line 1210) choice(a, size=None, replace=True, p=None) Generates a random sample from a given 1-D array .. versionadded:: 1.7.0 Parameters ----------- a : 1-D array-like or int If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if a were np.arange(a) size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. replace : boolean, optional Whether the sample is with or without replacement p : 1-D array-like, optional The probabilities associated with each entry in a. If not given the sample assumes a uniform distribution over all entries in a. Returns -------- samples : single item or ndarray The generated random samples Raises ------- ValueError If a is an int and less than zero, if a or p are not 1-dimensional, if a is an array-like of size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size See Also --------- randint, shuffle, permutation Examples --------- Generate a uniform random sample from np.arange(5) of size 3: >>> np.random.choice(5, 3) array([0, 3, 4]) >>> #This is equivalent to np.random.randint(0,5,3) Generate a non-uniform random sample from np.arange(5) of size 3: >>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0]) array([3, 3, 0]) Generate a uniform random sample from np.arange(5) of size 3 without replacement: >>> np.random.choice(5, 3, replace=False) array([3,1,0]) >>> #This is equivalent to np.random.permutation(np.arange(5))[:3] Generate a non-uniform random sample from np.arange(5) of size 3 without replacement: >>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0]) array([2, 3, 0]) Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance: >>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher'] >>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3]) array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], dtype='|S11') RandomState.choice (line 1028) bytes(length) Return random bytes. Parameters ---------- length : int Number of random bytes. Returns ------- out : str String of length `length`. Examples -------- >>> np.random.bytes(10) ' eh\x85\x022SZ\xbf\xa4' #random RandomState.bytes (line 999) randint(low, high=None, size=None, dtype='l') Return random integers from `low` (inclusive) to `high` (exclusive). Return random integers from the "discrete uniform" distribution of the specified dtype in the "half-open" interval [`low`, `high`). If `high` is None (the default), then results are from [0, `low`). Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution (unless ``high=None``, in which case this parameter is one above the *highest* such integer). high : int, optional If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if ``high=None``). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result. All dtypes are determined by their name, i.e., 'int64', 'int', etc, so byteorder is not available and a specific precision may have different C types depending on the platform. The default value is 'np.int'. .. versionadded:: 1.11.0 Returns ------- out : int or ndarray of ints `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. See Also -------- random.random_integers : similar to `randint`, only for the closed interval [`low`, `high`], and 1 is the lowest value if `high` is omitted. In particular, this other one is the one to use to generate uniformly distributed discrete non-integers. Examples -------- >>> np.random.randint(2, size=10) array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) >>> np.random.randint(1, size=10) array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) Generate a 2 x 4 array of ints between 0 and 4, inclusive: >>> np.random.randint(5, size=(2, 4)) array([[4, 0, 2, 1], [3, 2, 2, 0]]) RandomState.randint (line 905) tomaxint(size=None) Random integers between 0 and ``sys.maxint``, inclusive. Return a sample of uniformly distributed random integers in the interval [0, ``sys.maxint``]. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : ndarray Drawn samples, with shape `size`. See Also -------- randint : Uniform sampling over a given half-open interval of integers. random_integers : Uniform sampling over a given closed interval of integers. Examples -------- >>> RS = np.random.mtrand.RandomState() # need a RandomState object >>> RS.tomaxint((2,2,2)) array([[[1170048599, 1600360186], [ 739731006, 1947757578]], [[1871712945, 752307660], [1601631370, 1479324245]]]) >>> import sys >>> sys.maxint 2147483647 >>> RS.tomaxint((2,2,2)) < sys.maxint array([[[ True, True], [ True, True]], [[ True, True], [ True, True]]], dtype=bool) RandomState.tomaxint (line 858) random_sample(size=None) Return random floats in the half-open interval [0.0, 1.0). Results are from the "continuous uniform" distribution over the stated interval. To sample :math:`Unif[a, b), b > a` multiply the output of `random_sample` by `(b-a)` and add `a`:: (b - a) * random_sample() + a Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray of floats Array of random floats of shape `size` (unless ``size=None``, in which case a single float is returned). Examples -------- >>> np.random.random_sample() 0.47108547995356098 >>> type(np.random.random_sample()) >>> np.random.random_sample((5,)) array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428]) Three-by-two array of random numbers from [-5, 0): >>> 5 * np.random.random_sample((3, 2)) - 5 array([[-3.99149989, -0.52338984], [-2.99091858, -0.79479508], [-1.23204345, -1.75224494]]) RandomState.random_sample (line 814)noncentral_chisquarestandard_exponential/builddir/build/BUILD/numpy-1.13.3/numpy/random/mtrand/mtrand.pyx/builddir/build/BUILD/numpy-1.13.3/numpy/random/mtrand/randint_helpers.pxisum(pvals[:-1]) > 1.0covariance is not positive-semidefinite.check_valid must equal 'warn', 'raise', or 'ignore'mean and cov must have same lengthcov must be 2 dimensional and squaremean must be 1 dimensionalnumpy.dualp >= 1.0p <= 0.0ngood + nbad < nsamplensample < 1nbad < 0ngood < 0p > 1.0p < 0.0a <= 1.0lam value too large.lam value too largelam < 0n <= 0p is nanp > 1p < 0n < 0left == rightmode > rightleft > modescale <= 0.0mean <= 0.0scale <= 0mean <= 0scale < 0.0sigma < 0.0sigma < 0a < 0kappa < 0df <= 0nonc < 0dfnum <= 1dfden <= 0dfnum <= 0shape < 0b <= 0pnlfdbaTLa <= 0scale < 0This function is deprecated. Please call randint({low}, {high} + 1) insteadThis function is deprecated. 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