method
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).
Parameter of the distribution. Default is 0.
Parameter of the distribution. Must be non-negative. Default is 1.
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.
Drawn samples from the parameterized logistic distribution.
See also
probability density function, distribution or cumulative density function, etc.
The probability density for the Logistic distribution is
where \(\mu\) = location and \(s\) = 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.
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.
Weisstein, Eric W. “Logistic Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html
Wikipedia, “Logistic-distribution”, https://en.wikipedia.org/wiki/Logistic_distribution
Draw samples from the distribution:
# plot against distribution


method
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.
Mean value of the underlying normal distribution. Default is 0.
Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1.
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.
Drawn samples from the parameterized log-normal distribution.
See also
probability density function, distribution, cumulative density function, etc.
A variable x has a log-normal distribution if log(x) is normally distributed. The probability density function for the log-normal distribution is:
where \(\mu\) is the mean and \(\sigma\) 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.
Limpert, E., Stahel, W. A., and Abbt, M., “Log-normal Distributions across the Sciences: Keys and Clues,” BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme Values,” Basel: Birkhauser Verlag, 2001, pp. 31-32.
Draw samples from the distribution:
Display the histogram of the samples, along with the probability density function:
Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function.


method
Draw samples from a logarithmic series distribution.
Samples are drawn from a log series distribution with specified shape parameter, 0 < p < 1.
Shape parameter for the distribution. Must be in the range (0, 1).
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.
Drawn samples from the parameterized logarithmic series distribution.
See also
probability density function, distribution or cumulative density function, etc.
The probability mass function for the Log Series distribution is
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].
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).
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.
D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994.
Wikipedia, “Logarithmic distribution”, https://en.wikipedia.org/wiki/Logarithmic_distribution
Draw samples from the distribution:
# plot against distribution


method
Draw samples from a multinomial distribution.
The multinomial distribution is a multivariate generalization 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.
Number of experiments.
Probabilities of each of the p different outcomes with shape (k0, k1, ..., kn, p). Each element pvals[i,j,...,:] must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as sum(pvals[..., :-1], axis=-1) <= 1.0. Must have at least 1 dimension where pvals.shape[-1] > 0.
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn each with p elements. Default is None where the output size is determined by the broadcast shape of n and all by the final dimension of pvals, which is denoted as b=(b0, b1, ..., bq). If size is not None, then it must be compatible with the broadcast shape b. Specifically, size must have q or more elements and size[-(q-j):] must equal bj.
The drawn samples, of shape size, if provided. When size is provided, the output shape is size + (p,) If not specified, the shape is determined by the broadcast shape of n and pvals, (b0, b1, ..., bq) augmented with the dimension of the multinomial, p, so that that output shape is (b0, b1, ..., bq, p).
Each entry out[i,j,...,:] is a p-dimensional value drawn from the distribution.
Throw a dice 20 times:
It landed 4 times on 1, once on 2, etc.
Now, throw the dice 20 times, and 20 times again:
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.
Now, do one experiment throwing the dice 10 time, and 10 times again, and another throwing the dice 20 times, and 20 times again:
The first array shows the outcomes of throwing the dice 10 times, and the second shows the outcomes from throwing the dice 20 times.
A loaded die is more likely to land on number 6:
Simulate 10 throws of a 4-sided die and 20 throws of a 6-sided die
Generate categorical random variates from two categories where the first has 3 outcomes and the second has 2.
argmax(axis=-1) is then used to return the categories.
The same output dimension can be produced using broadcasting.
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:
not like:


method
Generate variates from a multivariate hypergeometric distribution.
The multivariate hypergeometric distribution is a generalization of the hypergeometric distribution.
Choose nsample items at random without replacement from a collection with N distinct types. N is the length of colors, and the values in colors are the number of occurrences of that type in the collection. The total number of items in the collection is sum(colors). Each random variate generated by this function is a vector of length N holding the counts of the different types that occurred in the nsample items.
The name colors comes from a common description of the distribution: it is the probability distribution of the number of marbles of each color selected without replacement from an urn containing marbles of different colors; colors[i] is the number of marbles in the urn with color i.
The number of each type of item in the collection from which a sample is drawn. The values in colors must be nonnegative. To avoid loss of precision in the algorithm, sum(colors) must be less than 10**9 when method is “marginals”.
The number of items selected. nsample must not be greater than sum(colors).
The number of variates to generate, either an integer or a tuple holding the shape of the array of variates. If the given size is, e.g., (k, m), then k * m variates are drawn, where one variate is a vector of length len(colors), and the return value has shape (k, m, len(colors)). If size is an integer, the output has shape (size, len(colors)). Default is None, in which case a single variate is returned as an array with shape (len(colors),).
Specify the algorithm that is used to generate the variates. Must be ‘count’ or ‘marginals’ (the default). See the Notes for a description of the methods.
Array of variates drawn from the multivariate hypergeometric distribution.
See also
Draw samples from the (univariate) hypergeometric distribution.
The two methods do not return the same sequence of variates.
The “count” algorithm is roughly equivalent to the following numpy code:
The “count” algorithm uses a temporary array of integers with length sum(colors).
The “marginals” algorithm generates a variate by using repeated calls to the univariate hypergeometric sampler. It is roughly equivalent to:
The default method is “marginals”. For some cases (e.g. when colors contains relatively small integers), the “count” method can be significantly faster than the “marginals” method. If performance of the algorithm is important, test the two methods with typical inputs to decide which works best.
New in version 1.18.0.


method
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.
Mean of the N-dimensional distribution.
Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling.
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.
Behavior when the covariance matrix is not positive semidefinite.
Tolerance when checking the singular values in covariance matrix. cov is cast to double before the check.
The cov input is used to compute a factor matrix A such that A @ A.T = cov. This argument is used to select the method used to compute the factor matrix A. The default method ‘svd’ is the slowest, while ‘cholesky’ is the fastest but less robust than the slowest method. The method eigh uses eigen decomposition to compute A and is faster than svd but slower than cholesky.
New in version 1.18.0.
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.
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, \(X = [x_1, x_2, ... x_N]\). The covariance matrix element \(C_{ij}\) is the covariance of \(x_i\) and \(x_j\). The element \(C_{ii}\) is the variance of \(x_i\) (i.e. its “spread”).
Instead of specifying the full covariance matrix, popular approximations include:
This geometrical property can be seen in two dimensions by plotting generated data-points:
Diagonal covariance means that points are oriented along x or y-axis:
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.
Papoulis, A., “Probability, Random Variables, and Stochastic Processes,” 3rd ed., New York: McGraw-Hill, 1991.
Duda, R. O., Hart, P. E., and Stork, D. G., “Pattern Classification,” 2nd ed., New York: Wiley, 2001.
We can use a different method other than the default to factorize cov:
The following is probably true, given that 0.6 is roughly twice the standard deviation:


method
Draw samples from a negative binomial distribution.
Samples are drawn from a negative binomial distribution with specified parameters, n successes and p probability of success where n is > 0 and p is in the interval (0, 1].
Parameter of the distribution, > 0.
Parameter of the distribution. Must satisfy 0 < p <= 1.
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.
Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached.
The probability mass function of the negative binomial distribution is
where \(n\) is the number of successes, \(p\) is the probability of success, \(N+n\) is the number of trials, and \(\Gamma\) is the gamma function. When \(n\) is an integer, \(\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}\), which is the more common form of this term in the the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last 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.
Weisstein, Eric W. “Negative Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html
Wikipedia, “Negative binomial distribution”, https://en.wikipedia.org/wiki/Negative_binomial_distribution
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.?


method
Draw samples from a noncentral chi-square distribution.
The noncentral \(\chi^2\) distribution is a generalization of the \(\chi^2\) distribution.
Degrees of freedom, must be > 0.
Changed in version 1.10.0: Earlier NumPy versions required dfnum > 1.
Non-centrality, must be non-negative.
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.
Drawn samples from the parameterized noncentral chi-square distribution.
The probability density function for the noncentral Chi-square distribution is
where \(Y_{q}\) is the Chi-square with q degrees of freedom.
Wikipedia, “Noncentral chi-squared distribution” https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution
Draw values from the distribution and plot the histogram
Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare.
Demonstrate how large values of non-centrality lead to a more symmetric distribution.


method
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.
Numerator degrees of freedom, must be > 0.
Changed in version 1.14.0: Earlier NumPy versions required dfnum > 1.
Denominator degrees of freedom, must be > 0.
Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0.
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.
Drawn samples from the parameterized noncentral Fisher distribution.
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.
Weisstein, Eric W. “Noncentral F-Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html
Wikipedia, “Noncentral F-distribution”, https://en.wikipedia.org/wiki/Noncentral_F-distribution
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.


method
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].
Mean (“centre”) of the distribution.
Standard deviation (spread or “width”) of the distribution. Must be non-negative.
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.
Drawn samples from the parameterized normal distribution.
See also
probability density function, distribution or cumulative density function, etc.
The probability density for the Gaussian distribution is
where \(\mu\) is the mean and \(\sigma\) the standard deviation. The square of the standard deviation, \(\sigma^2\), 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 \(x + \sigma\) and \(x - \sigma\) [2]). This implies that normal is more likely to return samples lying close to the mean, rather than those far away.
Wikipedia, “Normal distribution”, https://en.wikipedia.org/wiki/Normal_distribution
P. R. Peebles Jr., “Central Limit Theorem” in “Probability, Random Variables and Random Signal Principles”, 4th ed., 2001, pp. 51, 51, 125.
Draw samples from the distribution:
Verify the mean and the variance:
Display the histogram of the samples, along with the probability density function:
Two-by-four array of samples from N(3, 6.25):


method
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.
Shape of the distribution. Must be positive.
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.
Drawn samples from the parameterized Pareto distribution.
See also
probability density function, distribution or cumulative density function, etc.
probability density function, distribution or cumulative density function, etc.
The probability density for the Pareto distribution is
where \(a\) is the shape and \(m\) 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.
Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects.
Pareto, V. (1896). Course of Political Economy. Lausanne.
Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30.
Wikipedia, “Pareto distribution”, https://en.wikipedia.org/wiki/Pareto_distribution
Draw samples from the distribution:
Display the histogram of the samples, along with the probability density function:


method
Randomly permute a sequence, or return a permuted range.
If x is an integer, randomly permute np.arange(x). If x is an array, make a copy and shuffle the elements randomly.
The axis which x is shuffled along. Default is 0.
Permuted sequence or array range.


method
Randomly permute x along axis axis.
Unlike shuffle, each slice along the given axis is shuffled independently of the others.
Array to be shuffled.
Slices of x in this axis are shuffled. Each slice is shuffled independently of the others. If axis is None, the flattened array is shuffled.
If given, this is the destinaton of the shuffled array. If out is None, a shuffled copy of the array is returned.
If out is None, a shuffled copy of x is returned. Otherwise, the shuffled array is stored in out, and out is returned
See also
Create a numpy.random.Generator instance:
Create a test array:
Shuffle the rows of x:
x has not been modified:
To shuffle the rows of x in-place, pass x as the out parameter:
Note that when the out parameter is given, the return value is out:


method
Draw samples from a Poisson distribution.
The Poisson distribution is the limit of the binomial distribution for large N.
Expected number of events occurring in a fixed-time interval, must be >= 0. A sequence must be broadcastable over the requested size.
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.
Drawn samples from the parameterized Poisson distribution.
The Poisson distribution
For events with an expected separation \(\lambda\) the Poisson distribution \(f(k; \lambda)\) describes the probability of \(k\) events occurring within the observed interval \(\lambda\).
Because the output is limited to the range of the C int64 type, a ValueError is raised when lam is within 10 sigma of the maximum representable value.
Weisstein, Eric W. “Poisson Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html
Wikipedia, “Poisson distribution”, https://en.wikipedia.org/wiki/Poisson_distribution
Draw samples from the distribution:
Display histogram of the sample:
Draw each 100 values for lambda 100 and 500:


method
Draws samples in [0, 1] from a power distribution with positive exponent a - 1.
Also known as the power function distribution.
Parameter of the distribution. Must be non-negative.
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.
Drawn samples from the parameterized power distribution.
If a <= 0.
The probability density function is
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.
Christian Kleiber, Samuel Kotz, “Statistical size distributions in economics and actuarial sciences”, Wiley, 2003.
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. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf
Draw samples from the distribution:
Display the histogram of the samples, along with the probability density function:
Compare the power function distribution to the inverse of the Pareto.


method
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 \(Unif[a, b), b > a\) multiply the output of random by (b-a) and add a:
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.
Desired dtype of the result, only float64 and float32 are supported. Byteorder must be native. The default value is np.float64.
Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values.
Array of random floats of shape size (unless size=None, in which case a single float is returned).
Three-by-two array of random numbers from [-5, 0):


method
Draw samples from a Rayleigh distribution.
The \(\chi\) and Weibull distributions are generalizations of the Rayleigh.
Scale, also equals the mode. Must be non-negative. Default is 1.
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.
Drawn samples from the parameterized Rayleigh distribution.
The probability density function for the Rayleigh distribution is
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.
Brighton Webs Ltd., “Rayleigh Distribution,” https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp
Wikipedia, “Rayleigh distribution” https://en.wikipedia.org/wiki/Rayleigh_distribution
Draw values from the distribution and plot the histogram
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?
The percentage of waves larger than 3 meters is:


method
Modify an array or sequence in-place by shuffling its contents.
The order of sub-arrays is changed but their contents remains the same.
The array, list or mutable sequence to be shuffled.
The axis which x is shuffled along. Default is 0. It is only supported on ndarray objects.


method
Draw samples from a standard Cauchy distribution with mode = 0.
Also known as the Lorentz distribution.
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.
The drawn samples.
The probability density function for the full Cauchy distribution is
and the Standard Cauchy distribution just sets \(x_0=0\) and \(\gamma=1\)
The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis.
When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails.
NIST/SEMATECH e-Handbook of Statistical Methods, “Cauchy Distribution”, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
Weisstein, Eric W. “Cauchy Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html
Wikipedia, “Cauchy distribution” https://en.wikipedia.org/wiki/Cauchy_distribution
Draw samples and plot the distribution:


method
Draw samples from the standard exponential distribution.
standard_exponential is identical to the exponential distribution with a scale parameter of 1.
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.
Desired dtype of the result, only float64 and float32 are supported. Byteorder must be native. The default value is np.float64.
Either ‘inv’ or ‘zig’. ‘inv’ uses the default inverse CDF method. ‘zig’ uses the much faster Ziggurat method of Marsaglia and Tsang.
Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values.
Drawn samples.
Output a 3x8000 array:


method
Draw samples from a standard Gamma distribution.
Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale=1.
Parameter, must be non-negative.
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.
Desired dtype of the result, only float64 and float32 are supported. Byteorder must be native. The default value is np.float64.
Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values.
Drawn samples from the parameterized standard gamma distribution.
See also
probability density function, distribution or cumulative density function, etc.
The probability density for the Gamma distribution is
where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) 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.
Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html
Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution
Draw samples from the distribution:
Display the histogram of the samples, along with the probability density function:


method
Draw samples from a standard Normal distribution (mean=0, stdev=1).
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.
Desired dtype of the result, only float64 and float32 are supported. Byteorder must be native. The default value is np.float64.
Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values.
A floating-point array of shape size of drawn samples, or a single sample if size was not specified.
See also
Equivalent function with additional loc and scale arguments for setting the mean and standard deviation.
For random samples from \(N(\mu, \sigma^2)\), use one of:
Two-by-four array of samples from \(N(3, 6.25)\):


method
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).
Degrees of freedom, must be > 0.
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.
Drawn samples from the parameterized standard Student’s t distribution.
The probability density function for the t distribution is
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.
Dalgaard, Peter, “Introductory Statistics With R”, Springer, 2002.
Wikipedia, “Student’s t-distribution” https://en.wikipedia.org/wiki/Student’s_t-distribution
From Dalgaard page 83 [1], suppose the daily energy intake for 11 women in kilojoules (kJ) is:
Does their energy intake deviate systematically from the recommended value of 7725 kJ? Our null hypothesis will be the absence of deviation, and the alternate hypothesis will be the presence of an effect that could be either positive or negative, hence making our test 2-tailed.
Because we are estimating the mean and we have N=11 values in our sample, we have N-1=10 degrees of freedom. We set our significance level to 95% and compute the t statistic using the empirical mean and empirical standard deviation of our intake. We use a ddof of 1 to base the computation of our empirical standard deviation on an unbiased estimate of the variance (note: the final estimate is not unbiased due to the concave nature of the square root).
We draw 1000000 samples from Student’s t distribution with the adequate degrees of freedom.
Does our t statistic land in one of the two critical regions found at both tails of the distribution?
The probability value for this 2-tailed test is about 1.83%, which is lower than the 5% pre-determined significance threshold.
Therefore, the probability of observing values as extreme as our intake conditionally on the null hypothesis being true is too low, and we reject the null hypothesis of no deviation.


method
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.
Lower limit.
The value where the peak of the distribution occurs. The value must fulfill the condition left <= mode <= right.
Upper limit, must be larger than left.
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.
Drawn samples from the parameterized triangular distribution.
The probability density function for the triangular distribution is
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.
Wikipedia, “Triangular distribution” https://en.wikipedia.org/wiki/Triangular_distribution
Draw values from the distribution and plot the histogram:


method
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.
Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0.
Upper boundary of the output interval. All values generated will be less than high. The high limit may be included in the returned array of floats due to floating-point rounding in the equation low + (high-low) * random_sample(). high - low must be non-negative. The default value is 1.0.
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.
Drawn samples from the parameterized uniform distribution.
See also
Discrete uniform distribution, yielding integers.
Floats uniformly distributed over [0, 1).
The probability density function of the uniform distribution is
anywhere within the interval [a, b), and zero elsewhere.
When high == low, values of low will be returned.
Draw samples from the distribution:
All values are within the given interval:
Display the histogram of the samples, along with the probability density function:


method
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.
Mode (“center”) of the distribution.
Dispersion of the distribution, has to be >=0.
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.
Drawn samples from the parameterized von Mises distribution.
See also
probability density function, distribution, or cumulative density function, etc.
The probability density for the von Mises distribution is
where \(\mu\) is the mode and \(\kappa\) the dispersion, and \(I_0(\kappa)\) 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.
Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.
von Mises, R., “Mathematical Theory of Probability and Statistics”, New York: Academic Press, 1964.
Draw samples from the distribution:
Display the histogram of the samples, along with the probability density function:


method
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.
Distribution mean, must be > 0.
Scale parameter, must be > 0.
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.
Drawn samples from the parameterized Wald distribution.
The probability density function for the Wald distribution is
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.
Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp
Chhikara, Raj S., and Folks, J. Leroy, “The Inverse Gaussian Distribution: Theory : Methodology, and Applications”, CRC Press, 1988.
Wikipedia, “Inverse Gaussian distribution” https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution
Draw values from the distribution and plot the histogram:


method
Draw samples from a Weibull distribution.
Draw samples from a 1-parameter Weibull distribution with the given shape parameter a.
Here, U is drawn from the uniform distribution over (0,1].
The more common 2-parameter Weibull, including a scale parameter \(\lambda\) is just \(X = \lambda(-ln(U))^{1/a}\).
Shape parameter of the distribution. Must be nonnegative.
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.
Drawn samples from the parameterized Weibull distribution.
See also
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
where \(a\) is the shape and \(\lambda\) the scale.
The function has its peak (the mode) at \(\lambda(\frac{a-1}{a})^{1/a}\).
When a = 1, the Weibull distribution reduces to the exponential distribution.
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.
Waloddi Weibull, “A Statistical Distribution Function of Wide Applicability”, Journal Of Applied Mechanics ASME Paper 1951.
Wikipedia, “Weibull distribution”, https://en.wikipedia.org/wiki/Weibull_distribution
Draw samples from the distribution:
Display the histogram of the samples, along with the probability density function:


method
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 discrete probability distribution that satisfies Zipf’s law: the frequency of an item is inversely proportional to its rank in a frequency table.
Distribution parameter. Must be greater than 1.
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.
Drawn samples from the parameterized Zipf distribution.
See also
probability density function, distribution, or cumulative density function, etc.
The probability density for the Zipf distribution is
for integers \(k \geq 1\), where \(\zeta\) 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.
Zipf, G. K., “Selected Studies of the Principle of Relative Frequency in Language,” Cambridge, MA: Harvard Univ. Press, 1932.
Draw samples from the distribution:
Display the histogram of the samples, along with the expected histogram based on the probability density function:
bincount provides a fast histogram for small integers.


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
where p is the probability of success of an individual trial.
Note
New code should use the geometric method of a default_rng() instance instead; please see the Quick Start.
The probability of success of an individual trial.
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.
Drawn samples from the parameterized geometric distribution.
See also
which should be used for new code.
Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35:
How many trials succeeded after a single run?


Return a tuple representing the internal state of the generator.
For more details, see set_state.
Flag indicating to return a legacy tuple state when the BitGenerator is MT19937, instead of a dict.
The returned tuple has the following items:
If legacy is False, or the BitGenerator is not MT19937, then state is returned as a dictionary.
See also
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.


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.
Note
New code should use the gumbel method of a default_rng() instance instead; please see the Quick Start.
The location of the mode of the distribution. Default is 0.
The scale parameter of the distribution. Default is 1. Must be non- negative.
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.
Drawn samples from the parameterized Gumbel distribution.
See also
which should be used for new code.
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
where \(\mu\) is the mode, a location parameter, and \(\beta\) 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 \(\mu + 0.57721\beta\) and a variance of \(\frac{\pi^2}{6}\beta^2\).
Gumbel, E. J., “Statistics of Extremes,” New York: Columbia University Press, 1958.
Reiss, R.-D. and Thomas, M., “Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields,” Basel: Birkhauser Verlag, 2001.
Draw samples from the distribution:
Display the histogram of the samples, along with the probability density function:
Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian:


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).
Note
New code should use the hypergeometric method of a default_rng() instance instead; please see the Quick Start.
Number of ways to make a good selection. Must be nonnegative.
Number of ways to make a bad selection. Must be nonnegative.
Number of items sampled. Must be at least 1 and at most ngood + nbad.
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.
Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size nsample taken from a set of ngood good items and nbad bad items.
See also
probability density function, distribution or cumulative density function, etc.
which should be used for new code.
The probability density for the Hypergeometric distribution is
where \(0 \le x \le n\) and \(n-b \le x \le g\)
for P(x) the probability of x good results in the drawn sample, g = ngood, b = nbad, and n = nsample.
Consider an urn with black and white marbles in it, ngood of them are 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.
Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.
Weisstein, Eric W. “Hypergeometric Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html
Wikipedia, “Hypergeometric distribution”, https://en.wikipedia.org/wiki/Hypergeometric_distribution
Draw samples from the distribution:
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?


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.
Note
New code should use the laplace method of a default_rng() instance instead; please see the Quick Start.
The position, \(\mu\), of the distribution peak. Default is 0.
\(\lambda\), the exponential decay. Default is 1. Must be non- negative.
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.
Drawn samples from the parameterized Laplace distribution.
See also
which should be used for new code.
It has the probability density function
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.
Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.
Kotz, Samuel, et. al. “The Laplace Distribution and Generalizations, ” Birkhauser, 2001.
Weisstein, Eric W. “Laplace Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html
Wikipedia, “Laplace distribution”, https://en.wikipedia.org/wiki/Laplace_distribution
Draw samples from the distribution
Display the histogram of the samples, along with the probability density function:
Plot Gaussian for comparison:


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).
Note
New code should use the logistic method of a default_rng() instance instead; please see the Quick Start.
Parameter of the distribution. Default is 0.
Parameter of the distribution. Must be non-negative. Default is 1.
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.
Drawn samples from the parameterized logistic distribution.
See also
probability density function, distribution or cumulative density function, etc.
which should be used for new code.
The probability density for the Logistic distribution is
where \(\mu\) = location and \(s\) = 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.
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.
Weisstein, Eric W. “Logistic Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html
Wikipedia, “Logistic-distribution”, https://en.wikipedia.org/wiki/Logistic_distribution
Draw samples from the distribution:
# plot against distribution


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.
Note
New code should use the lognormal method of a default_rng() instance instead; please see the Quick Start.
Mean value of the underlying normal distribution. Default is 0.
Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1.
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.
Drawn samples from the parameterized log-normal distribution.
See also
probability density function, distribution, cumulative density function, etc.
which should be used for new code.
A variable x has a log-normal distribution if log(x) is normally distributed. The probability density function for the log-normal distribution is:
where \(\mu\) is the mean and \(\sigma\) 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.
Limpert, E., Stahel, W. A., and Abbt, M., “Log-normal Distributions across the Sciences: Keys and Clues,” BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme Values,” Basel: Birkhauser Verlag, 2001, pp. 31-32.
Draw samples from the distribution:
Display the histogram of the samples, along with the probability density function:
Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function.


Draw samples from a logarithmic series distribution.
Samples are drawn from a log series distribution with specified shape parameter, 0 < p < 1.
Note
New code should use the logseries method of a default_rng() instance instead; please see the Quick Start.
Shape parameter for the distribution. Must be in the range (0, 1).
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.
Drawn samples from the parameterized logarithmic series distribution.
See also
probability density function, distribution or cumulative density function, etc.
which should be used for new code.
The probability density for the Log Series distribution is
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].
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).
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.
D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994.
Wikipedia, “Logarithmic distribution”, https://en.wikipedia.org/wiki/Logarithmic_distribution
Draw samples from the distribution:
# plot against distribution


attribute
CFFI interface
Named tuple containing CFFI wrapper


attribute
ctypes interface
Named tuple containing ctypes wrapper


method
Returns a new bit generator with the state jumped
The state of the returned big generator is jumped as-if 2**(128 * jumps) random numbers have been generated.
Number of times to jump the state of the bit generator returned
New instance of generator jumped iter times
The jump step is computed using a modified version of Matsumoto’s implementation of Horner’s method. The step polynomial is precomputed to perform 2**128 steps. The jumped state has been verified to match the state produced using Matsumoto’s original code.
Matsumoto, M, Generating multiple disjoint streams of pseudorandom number sequences. Accessed on: May 6, 2020. http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/JUMP/
Hiroshi Haramoto, Makoto Matsumoto, Takuji Nishimura, François Panneton, Pierre L’Ecuyer, “Efficient Jump Ahead for F2-Linear Random Number Generators”, INFORMS JOURNAL ON COMPUTING, Vol. 20, No. 3, Summer 2008, pp. 385-390.


attribute
Get or set the PRNG state
Dictionary containing the information required to describe the state of the PRNG


Draw samples from a multinomial distribution.
The multinomial distribution is a multivariate generalization 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.
Note
New code should use the multinomial method of a default_rng() instance instead; please see the Quick Start.
Number of experiments.
Probabilities of each of the p different outcomes. These must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as sum(pvals[:-1]) <= 1).
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.
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.
See also
which should be used for new code.
Throw a dice 20 times:
It landed 4 times on 1, once on 2, etc.
Now, throw the dice 20 times, and 20 times again:
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:
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:
not like:


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.
Note
New code should use the multivariate_normal method of a default_rng() instance instead; please see the Quick Start.
Mean of the N-dimensional distribution.
Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling.
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.
Behavior when the covariance matrix is not positive semidefinite.
Tolerance when checking the singular values in covariance matrix. cov is cast to double before the check.
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.
See also
which should be used for new code.
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, \(X = [x_1, x_2, ... x_N]\). The covariance matrix element \(C_{ij}\) is the covariance of \(x_i\) and \(x_j\). The element \(C_{ii}\) is the variance of \(x_i\) (i.e. its “spread”).
Instead of specifying the full covariance matrix, popular approximations include:
This geometrical property can be seen in two dimensions by plotting generated data-points:
Diagonal covariance means that points are oriented along x or y-axis:
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.
Papoulis, A., “Probability, Random Variables, and Stochastic Processes,” 3rd ed., New York: McGraw-Hill, 1991.
Duda, R. O., Hart, P. E., and Stork, D. G., “Pattern Classification,” 2nd ed., New York: Wiley, 2001.
The following is probably true, given that 0.6 is roughly twice the standard deviation:


Draw samples from a negative binomial distribution.
Samples are drawn from a negative binomial distribution with specified parameters, n successes and p probability of success where n is > 0 and p is in the interval [0, 1].
Note
New code should use the negative_binomial method of a default_rng() instance instead; please see the Quick Start.
Parameter of the distribution, > 0.
Parameter of the distribution, >= 0 and <=1.
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.
Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached.
See also
which should be used for new code.
The probability mass function of the negative binomial distribution is
where \(n\) is the number of successes, \(p\) is the probability of success, \(N+n\) is the number of trials, and \(\Gamma\) is the gamma function. When \(n\) is an integer, \(\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}\), which is the more common form of this term in the the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last 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.
Weisstein, Eric W. “Negative Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html
Wikipedia, “Negative binomial distribution”, https://en.wikipedia.org/wiki/Negative_binomial_distribution
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.?


Draw samples from a noncentral chi-square distribution.
The noncentral \(\chi^2\) distribution is a generalization of the \(\chi^2\) distribution.
Note
New code should use the noncentral_chisquare method of a default_rng() instance instead; please see the Quick Start.
Degrees of freedom, must be > 0.
Changed in version 1.10.0: Earlier NumPy versions required dfnum > 1.
Non-centrality, must be non-negative.
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.
Drawn samples from the parameterized noncentral chi-square distribution.
See also
which should be used for new code.
The probability density function for the noncentral Chi-square distribution is
where \(Y_{q}\) is the Chi-square with q degrees of freedom.
Wikipedia, “Noncentral chi-squared distribution” https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution
Draw values from the distribution and plot the histogram
Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare.
Demonstrate how large values of non-centrality lead to a more symmetric distribution.


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.
Note
New code should use the noncentral_f method of a default_rng() instance instead; please see the Quick Start.
Numerator degrees of freedom, must be > 0.
Changed in version 1.14.0: Earlier NumPy versions required dfnum > 1.
Denominator degrees of freedom, must be > 0.
Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0.
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.
Drawn samples from the parameterized noncentral Fisher distribution.
See also
which should be used for new code.
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.
Weisstein, Eric W. “Noncentral F-Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html
Wikipedia, “Noncentral F-distribution”, https://en.wikipedia.org/wiki/Noncentral_F-distribution
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.


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].
Note
New code should use the normal method of a default_rng() instance instead; please see the Quick Start.
Mean (“centre”) of the distribution.
Standard deviation (spread or “width”) of the distribution. Must be non-negative.
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.
Drawn samples from the parameterized normal distribution.
See also
probability density function, distribution or cumulative density function, etc.
which should be used for new code.
The probability density for the Gaussian distribution is
where \(\mu\) is the mean and \(\sigma\) the standard deviation. The square of the standard deviation, \(\sigma^2\), 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 \(x + \sigma\) and \(x - \sigma\) [2]). This implies that normal is more likely to return samples lying close to the mean, rather than those far away.
Wikipedia, “Normal distribution”, https://en.wikipedia.org/wiki/Normal_distribution
P. R. Peebles Jr., “Central Limit Theorem” in “Probability, Random Variables and Random Signal Principles”, 4th ed., 2001, pp. 51, 51, 125.
Draw samples from the distribution:
Verify the mean and the variance:
Display the histogram of the samples, along with the probability density function:
Two-by-four array of samples from N(3, 6.25):


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.
Note
New code should use the pareto method of a default_rng() instance instead; please see the Quick Start.
Shape of the distribution. Must be positive.
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.
Drawn samples from the parameterized Pareto distribution.
See also
probability density function, distribution or cumulative density function, etc.
probability density function, distribution or cumulative density function, etc.
which should be used for new code.
The probability density for the Pareto distribution is
where \(a\) is the shape and \(m\) 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.
Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects.
Pareto, V. (1896). Course of Political Economy. Lausanne.
Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30.
Wikipedia, “Pareto distribution”, https://en.wikipedia.org/wiki/Pareto_distribution
Draw samples from the distribution:
Display the histogram of the samples, along with the probability density function:


method
Advance the underlying RNG as-if delta draws have occurred.
Number of draws to advance the RNG. Must be less than the size state variable in the underlying RNG.
RNG advanced delta steps
Advancing a RNG updates the underlying RNG state as-if a given number of calls to the underlying RNG have been made. In general there is not a one-to-one relationship between the number output random values from a particular distribution and the number of draws from the core RNG. This occurs for two reasons:
Advancing the RNG state resets any pre-computed random numbers. This is required to ensure exact reproducibility.


method
Returns a new bit generator with the state jumped.
Jumps the state as-if jumps * 210306068529402873165736369884012333109 random numbers have been generated.
Number of times to jump the state of the bit generator returned
New instance of generator jumped iter times
The step size is phi-1 when multiplied by 2**128 where phi is the golden ratio.


Randomly permute a sequence, or return a permuted range.
If x is a multi-dimensional array, it is only shuffled along its first index.
Note
New code should use the permutation method of a default_rng() instance instead; please see the Quick Start.
If x is an integer, randomly permute np.arange(x). If x is an array, make a copy and shuffle the elements randomly.
Permuted sequence or array range.
See also
which should be used for new code.


method
Returns a new bit generator with the state jumped
The state of the returned big generator is jumped as-if 2**(128 * jumps) random numbers have been generated.
Number of times to jump the state of the bit generator returned
New instance of generator jumped iter times


Draw samples from a Poisson distribution.
The Poisson distribution is the limit of the binomial distribution for large N.
Note
New code should use the poisson method of a default_rng() instance instead; please see the Quick Start.
Expected number of events occurring in a fixed-time interval, must be >= 0. A sequence must be broadcastable over the requested size.
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.
Drawn samples from the parameterized Poisson distribution.
See also
which should be used for new code.
The Poisson distribution
For events with an expected separation \(\lambda\) the Poisson distribution \(f(k; \lambda)\) describes the probability of \(k\) events occurring within the observed interval \(\lambda\).
Because the output is limited to the range of the C int64 type, a ValueError is raised when lam is within 10 sigma of the maximum representable value.
Weisstein, Eric W. “Poisson Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html
Wikipedia, “Poisson distribution”, https://en.wikipedia.org/wiki/Poisson_distribution
Draw samples from the distribution:
Display histogram of the sample:
Draw each 100 values for lambda 100 and 500:


Draws samples in [0, 1] from a power distribution with positive exponent a - 1.
Also known as the power function distribution.
Note
New code should use the power method of a default_rng() instance instead; please see the Quick Start.
Parameter of the distribution. Must be non-negative.
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.
Drawn samples from the parameterized power distribution.
If a <= 0.
See also
which should be used for new code.
The probability density function is
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.
Christian Kleiber, Samuel Kotz, “Statistical size distributions in economics and actuarial sciences”, Wiley, 2003.
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. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf
Draw samples from the distribution:
Display the histogram of the samples, along with the probability density function:
Compare the power function distribution to the inverse of the Pareto.


Random values in a given shape.
Note
This is a convenience function for users porting code from Matlab, and wraps random_sample. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.
Create an array of the given shape and populate it with random samples from a uniform distribution over [0, 1).
The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.
Random values.
See also


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).
Note
New code should use the integers method of a default_rng() instance instead; please see the Quick Start.
Lowest (signed) integers to be drawn from the distribution (unless high=None, in which case this parameter is one above the highest such integer).
If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None). If array-like, must contain integer values
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.
Desired dtype of the result. Byteorder must be native. The default value is int.
New in version 1.11.0.
size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.
See also
similar to randint, only for the closed interval [low, high], and 1 is the lowest value if high is omitted.
which should be used for new code.
Generate a 2 x 4 array of ints between 0 and 4, inclusive:
Generate a 1 x 3 array with 3 different upper bounds
Generate a 1 by 3 array with 3 different lower bounds
Generate a 2 by 4 array using broadcasting with dtype of uint8


Return a sample (or samples) from the “standard normal” distribution.
Note
This is a convenience function for users porting code from Matlab, and wraps standard_normal. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.
Note
New code should use the standard_normal method of a default_rng() instance instead; please see the Quick Start.
If positive int_like 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. A single float randomly sampled from the distribution is returned if no argument is provided.
The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.
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
Similar, but takes a tuple as its argument.
Also accepts mu and sigma arguments.
which should be used for new code.
For random samples from \(N(\mu, \sigma^2)\), use:
sigma * np.random.randn(...) + mu
Two-by-four array of samples from N(3, 6.25):


Return random floats in the half-open interval [0.0, 1.0). Alias for random_sample to ease forward-porting to the new random API.


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 integer type and its precision is platform dependent.
This function has been deprecated. Use randint instead.
Deprecated since version 1.11.0.
Lowest (signed) integer to be drawn from the distribution (unless high=None, in which case this parameter is the highest such integer).
If provided, the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None).
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.
size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.
See also
Similar to random_integers, only for the half-open interval [low, high), and 0 is the lowest value if high is omitted.
To sample from N evenly spaced floating-point numbers between a and b, use:
Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (i.e., from the set \({0, 5/8, 10/8, 15/8, 20/8}\)):
Roll two six sided dice 1000 times and sum the results:
Display results as a histogram:


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 \(Unif[a, b), b > a\) multiply the output of random_sample by (b-a) and add a:
Note
New code should use the random method of a default_rng() instance instead; please see the Quick Start.
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.
Array of random floats of shape size (unless size=None, in which case a single float is returned).
See also
which should be used for new code.
Three-by-two array of random numbers from [-5, 0):


method
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
where the normalization, B, is the beta function,
It is often seen in Bayesian inference and order statistics.
Note
New code should use the beta method of a default_rng() instance instead; please see the Quick Start.
Alpha, positive (>0).
Beta, positive (>0).
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.
Drawn samples from the parameterized beta distribution.
See also
which should be used for new code.


method
Draw samples from a binomial distribution.
Samples are drawn from a binomial distribution with specified parameters, n trials and p probability of success where n an integer >= 0 and p is in the interval [0,1]. (n may be input as a float, but it is truncated to an integer in use)
Note
New code should use the binomial method of a default_rng() instance instead; please see the Quick Start.
Parameter of the distribution, >= 0. Floats are also accepted, but they will be truncated to integers.
Parameter of the distribution, >= 0 and <=1.
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.
Drawn samples from the parameterized binomial distribution, where each sample is equal to the number of successes over the n trials.
See also
probability density function, distribution or cumulative density function, etc.
which should be used for new code.
The probability density for the binomial distribution is
where \(n\) is the number of trials, \(p\) is the probability of success, and \(N\) is the number of successes.
When estimating the standard error of a proportion in a population by using a random sample, the normal distribution works well unless the product p*n <=5, where p = population proportion estimate, and n = number of samples, in which case the binomial distribution is used instead. For example, a sample of 15 people shows 4 who are left handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4, so the binomial distribution should be used in this case.
Dalgaard, Peter, “Introductory Statistics with R”, Springer-Verlag, 2002.
Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.
Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.
Weisstein, Eric W. “Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/BinomialDistribution.html
Wikipedia, “Binomial distribution”, https://en.wikipedia.org/wiki/Binomial_distribution
Draw samples from the distribution:
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.


method
Return random bytes.
Note
New code should use the bytes method of a default_rng() instance instead; please see the Quick Start.
Number of random bytes.
String of length length.
See also
which should be used for new code.


method
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.
Note
New code should use the chisquare method of a default_rng() instance instead; please see the Quick Start.
Number of degrees of freedom, must be > 0.
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.
Drawn samples from the parameterized chi-square distribution.
When df <= 0 or when an inappropriate size (e.g. size=-1) is given.
See also
which should be used for new code.
The variable obtained by summing the squares of df independent, standard normally distributed random variables:
is chi-square distributed, denoted
The probability density function of the chi-squared distribution is
where \(\Gamma\) is the gamma function,
NIST “Engineering Statistics Handbook” https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm


method
Generates a random sample from a given 1-D array
New in version 1.7.0.
Note
New code should use the choice method of a default_rng() instance instead; please see the Quick Start.
If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if it were np.arange(a)
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.
Whether the sample is with or without replacement. Default is True, meaning that a value of a can be selected multiple times.
The probabilities associated with each entry in a. If not given, the sample assumes a uniform distribution over all entries in a.
The generated random samples
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
which should be used in new code
Setting user-specified probabilities through p uses a more general but less efficient sampler than the default. The general sampler produces a different sample than the optimized sampler even if each element of p is 1 / len(a).
Sampling random rows from a 2-D array is not possible with this function, but is possible with Generator.choice through its axis keyword.
Generate a uniform random sample from np.arange(5) of size 3:
Generate a non-uniform random sample from np.arange(5) of size 3:
Generate a uniform random sample from np.arange(5) of size 3 without replacement:
Generate a non-uniform random sample from np.arange(5) of size 3 without replacement:
Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance:


method
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. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference.
Note
New code should use the dirichlet method of a default_rng() instance instead; please see the Quick Start.
Parameter of the distribution (length k for sample of length k).
Output shape. If the given shape is, e.g., (m, n), then m * n * k samples are drawn. Default is None, in which case a vector of length k is returned.
The drawn samples, of shape (size, k).
If any value in alpha is less than or equal to zero
See also
which should be used for new code.
The Dirichlet distribution is a distribution over vectors \(x\) that fulfil the conditions \(x_i>0\) and \(\sum_{i=1}^k x_i = 1\).
The probability density function \(p\) of a Dirichlet-distributed random vector \(X\) is proportional to
where \(\alpha\) is a vector containing the positive concentration parameters.
The method uses the following property for computation: let \(Y\) be a random vector which has components that follow a standard gamma distribution, then \(X = \frac{1}{\sum_{i=1}^k{Y_i}} Y\) is Dirichlet-distributed
David McKay, “Information Theory, Inference and Learning Algorithms,” chapter 23, http://www.inference.org.uk/mackay/itila/
Wikipedia, “Dirichlet distribution”, https://en.wikipedia.org/wiki/Dirichlet_distribution
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.


method
Draw samples from an exponential distribution.
Its probability density function is
for x > 0 and 0 elsewhere. \(\beta\) is the scale parameter, which is the inverse of the rate parameter \(\lambda = 1/\beta\). 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].
Note
New code should use the exponential method of a default_rng() instance instead; please see the Quick Start.
The scale parameter, \(\beta = 1/\lambda\). Must be non-negative.
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.
Drawn samples from the parameterized exponential distribution.
See also
which should be used for new code.
Peyton Z. Peebles Jr., “Probability, Random Variables and Random Signal Principles”, 4th ed, 2001, p. 57.
Wikipedia, “Poisson process”, https://en.wikipedia.org/wiki/Poisson_process
Wikipedia, “Exponential distribution”, https://en.wikipedia.org/wiki/Exponential_distribution


method
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 must 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.
Note
New code should use the f method of a default_rng() instance instead; please see the Quick Start.
Degrees of freedom in numerator, must be > 0.
Degrees of freedom in denominator, must be > 0.
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.
Drawn samples from the parameterized Fisher distribution.
See also
probability density function, distribution or cumulative density function, etc.
which should be used for new code.
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.
Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.
Wikipedia, “F-distribution”, https://en.wikipedia.org/wiki/F-distribution
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:
The lower bound for the top 1% of the samples is :
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.


method
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.
Note
New code should use the gamma method of a default_rng() instance instead; please see the Quick Start.
The shape of the gamma distribution. Must be non-negative.
The scale of the gamma distribution. Must be non-negative. Default is equal to 1.
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.
Drawn samples from the parameterized gamma distribution.
See also
probability density function, distribution or cumulative density function, etc.
which should be used for new code.
The probability density for the Gamma distribution is
where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) 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.
Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html
Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution
Draw samples from the distribution:
Display the histogram of the samples, along with the probability density function:


method
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
where p is the probability of success of an individual trial.
Note
New code should use the geometric method of a default_rng() instance instead; please see the Quick Start.
The probability of success of an individual trial.
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.
Drawn samples from the parameterized geometric distribution.
See also
which should be used for new code.
Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35:
How many trials succeeded after a single run?


method
Return a tuple representing the internal state of the generator.
For more details, see set_state.
Flag indicating to return a legacy tuple state when the BitGenerator is MT19937, instead of a dict.
The returned tuple has the following items:
If legacy is False, or the BitGenerator is not MT19937, then state is returned as a dictionary.
See also
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.