method
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:


method
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?


method
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:


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).
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


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.
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.


method
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


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.
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:


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.
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:


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].
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.?


method
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.


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.
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.


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].
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):


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.
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
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
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:


method
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.


method
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


method
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


method
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):


method
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:


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_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 Rayleigh distribution.
The \(\chi\) and Weibull distributions are generalizations of the Rayleigh.
Note
New code should use the rayleigh method of a default_rng() instance instead; please see the Quick Start.
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.
See also
which should be used for new code.
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
Reseed a legacy MT19937 BitGenerator
This is a convenience, legacy function.
The best practice is to not reseed a BitGenerator, rather to recreate a new one. This method is here for legacy reasons. This example demonstrates best practice.


method
Set the internal state of the generator from a tuple.
For use if one has reason to manually (re-)set the internal state of the bit generator used by the RandomState instance. By default, RandomState uses the “Mersenne Twister”[1] pseudo-random number generating algorithm.
The state tuple has the following items:
If state is a dictionary, it is directly set using the BitGenerators state property.
Returns ‘None’ on success.
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.
For backwards compatibility, the form (str, array of 624 uints, int) is also accepted although it is missing some information about the cached Gaussian value: state = ('MT19937', keys, pos).
M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. on Modeling and Computer Simulation, Vol. 8, No. 1, pp. 3-30, Jan. 1998.


method
Modify a sequence in-place by shuffling its contents.
This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same.
Note
New code should use the shuffle method of a default_rng() instance instead; please see the Quick Start.
The array, list or mutable sequence to be shuffled.
See also
which should be used for new code.
Multi-dimensional arrays are only shuffled along the first axis:


method
Draw samples from a standard Cauchy distribution with mode = 0.
Also known as the Lorentz distribution.
Note
New code should use the standard_cauchy 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.
The drawn samples.
See also
which should be used for new code.
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.
Note
New code should use the standard_exponential 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.
Drawn samples.
See also
which should be used for new code.
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.
Note
New code should use the standard_gamma method of a default_rng() instance instead; please see the Quick Start.
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.
Drawn samples from the parameterized standard 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 a standard Normal distribution (mean=0, stdev=1).
Note
New code should use the standard_normal 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.
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.
which should be used for new code.
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).
Note
New code should use the standard_t method of a default_rng() instance instead; please see the Quick Start.
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.
See also
which should be used for new code.
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.
Note
New code should use the triangular method of a default_rng() instance instead; please see the Quick Start.
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.
See also
which should be used for new code.
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.
Note
New code should use the uniform method of a default_rng() instance instead; please see the Quick Start.
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 or equal to 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(). 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.
Discrete uniform distribution over the closed interval [low, high].
Floats uniformly distributed over [0, 1).
Alias for random_sample.
Convenience function that accepts dimensions as input, e.g., rand(2,2) would generate a 2-by-2 array of floats, uniformly distributed over [0, 1).
which should be used for new code.
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. If high < low, the results are officially undefined and may eventually raise an error, i.e. do not rely on this function to behave when passed arguments satisfying that inequality condition. 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(). For example:
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.
Note
New code should use the vonmises method of a default_rng() instance instead; please see the Quick Start.
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.
which should be used for new code.
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.
Note
New code should use the wald method of a default_rng() instance instead; please see the Quick Start.
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.
See also
which should be used for new code.
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 \(X = \lambda(-ln(U))^{1/a}\).
Note
New code should use the weibull method of a default_rng() instance instead; please see the Quick Start.
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
which should be used for new code.
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.
Note
New code should use the zipf method of a default_rng() instance instead; please see the Quick Start.
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.
which should be used for new code.
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.


This is an alias of random_sample. See random_sample for the complete documentation.


Draw samples from a Rayleigh distribution.
The \(\chi\) and Weibull distributions are generalizations of the Rayleigh.
Note
New code should use the rayleigh method of a default_rng() instance instead; please see the Quick Start.
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.
See also
which should be used for new code.
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:


Reseed a legacy MT19937 BitGenerator
This is a convenience, legacy function.
The best practice is to not reseed a BitGenerator, rather to recreate a new one. This method is here for legacy reasons. This example demonstrates best practice.


method
Return the requested number of words for PRNG seeding.
A BitGenerator should call this method in its constructor with an appropriate n_words parameter to properly seed itself.
The size of each word. This should only be either uint32 or uint64. Strings (‘uint32’, ‘uint64’) are fine. Note that requesting uint64 will draw twice as many bits as uint32 for the same n_words. This is a convenience for BitGenerator`s that express their states as `uint64 arrays.


method
Spawn a number of child SeedSequence s by extending the spawn_key.


Set the internal state of the generator from a tuple.
For use if one has reason to manually (re-)set the internal state of the bit generator used by the RandomState instance. By default, RandomState uses the “Mersenne Twister”[1] pseudo-random number generating algorithm.
The state tuple has the following items:
If state is a dictionary, it is directly set using the BitGenerators state property.
Returns ‘None’ on success.
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.
For backwards compatibility, the form (str, array of 624 uints, int) is also accepted although it is missing some information about the cached Gaussian value: state = ('MT19937', keys, pos).
M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. on Modeling and Computer Simulation, Vol. 8, No. 1, pp. 3-30, Jan. 1998.


attribute
CFFI interface
Named tuple containing CFFI wrapper


attribute
ctypes interface
Named tuple containing ctypes wrapper


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


Modify a sequence in-place by shuffling its contents.
This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same.
Note
New code should use the shuffle method of a default_rng() instance instead; please see the Quick Start.
The array, list or mutable sequence to be shuffled.
See also
which should be used for new code.
Multi-dimensional arrays are only shuffled along the first axis:


Draw samples from a standard Cauchy distribution with mode = 0.
Also known as the Lorentz distribution.
Note
New code should use the standard_cauchy 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.
The drawn samples.
See also
which should be used for new code.
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:


Draw samples from the standard exponential distribution.
standard_exponential is identical to the exponential distribution with a scale parameter of 1.
Note
New code should use the standard_exponential 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.
Drawn samples.
See also
which should be used for new code.
Output a 3x8000 array:


Draw samples from a standard Gamma distribution.
Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale=1.
Note
New code should use the standard_gamma method of a default_rng() instance instead; please see the Quick Start.
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.
Drawn samples from the parameterized standard 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:


Draw samples from a standard Normal distribution (mean=0, stdev=1).
Note
New code should use the standard_normal 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.
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.
which should be used for new code.
For random samples from \(N(\mu, \sigma^2)\), use one of:
Two-by-four array of samples from \(N(3, 6.25)\):


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).
Note
New code should use the standard_t method of a default_rng() instance instead; please see the Quick Start.
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.
See also
which should be used for new code.
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.


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.
Note
New code should use the triangular method of a default_rng() instance instead; please see the Quick Start.
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.
See also
which should be used for new code.
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:


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.
Note
New code should use the uniform method of a default_rng() instance instead; please see the Quick Start.
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 or equal to 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(). 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.
Discrete uniform distribution over the closed interval [low, high].
Floats uniformly distributed over [0, 1).
Alias for random_sample.
Convenience function that accepts dimensions as input, e.g., rand(2,2) would generate a 2-by-2 array of floats, uniformly distributed over [0, 1).
which should be used for new code.
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. If high < low, the results are officially undefined and may eventually raise an error, i.e. do not rely on this function to behave when passed arguments satisfying that inequality condition. 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(). For example:
Draw samples from the distribution:
All values are within the given interval:
Display the histogram of the samples, along with the probability density function:


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.
Note
New code should use the vonmises method of a default_rng() instance instead; please see the Quick Start.
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.
which should be used for new code.
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:


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.
Note
New code should use the wald method of a default_rng() instance instead; please see the Quick Start.
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.
See also
which should be used for new code.
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:


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 \(X = \lambda(-ln(U))^{1/a}\).
Note
New code should use the weibull method of a default_rng() instance instead; please see the Quick Start.
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
which should be used for new code.
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:


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.
Note
New code should use the zipf method of a default_rng() instance instead; please see the Quick Start.
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.
which should be used for new code.
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.


method
Returns True if all elements evaluate to True.
Refer to numpy.all for full documentation.
See also
equivalent function


method
Returns True if any of the elements of a evaluate to True.
Refer to numpy.any for full documentation.
See also
equivalent function


method
Return indices of the maximum values along the given axis.
Refer to numpy.argmax for full documentation.
See also
equivalent function


method
Return indices of the minimum values along the given axis.
Refer to numpy.argmin for detailed documentation.
See also
equivalent function


method
Returns the indices that would partition this array.
Refer to numpy.argpartition for full documentation.
New in version 1.8.0.
See also
equivalent function


method
Returns the indices that would sort this array.
Refer to numpy.argsort for full documentation.
See also
equivalent function


method
Copy of the array, cast to a specified type.
Typecode or data-type to which the array is cast.
Controls the memory layout order of the result. ‘C’ means C order, ‘F’ means Fortran order, ‘A’ means ‘F’ order if all the arrays are Fortran contiguous, ‘C’ order otherwise, and ‘K’ means as close to the order the array elements appear in memory as possible. Default is ‘K’.
Controls what kind of data casting may occur. Defaults to ‘unsafe’ for backwards compatibility.
If True, then sub-classes will be passed-through (default), otherwise the returned array will be forced to be a base-class array.
By default, astype always returns a newly allocated array. If this is set to false, and the dtype, order, and subok requirements are satisfied, the input array is returned instead of a copy.
Unless copy is False and the other conditions for returning the input array are satisfied (see description for copy input parameter), arr_t is a new array of the same shape as the input array, with dtype, order given by dtype, order.
When casting from complex to float or int. To avoid this, one should use a.real.astype(t).
Changed in version 1.17.0: Casting between a simple data type and a structured one is possible only for “unsafe” casting. Casting to multiple fields is allowed, but casting from multiple fields is not.
Changed in version 1.9.0: Casting from numeric to string types in ‘safe’ casting mode requires that the string dtype length is long enough to store the max integer/float value converted.


attribute
Base object if memory is from some other object.
The base of an array that owns its memory is None:
Slicing creates a view, whose memory is shared with x:


method
Swap the bytes of the array elements
Toggle between low-endian and big-endian data representation by returning a byteswapped array, optionally swapped in-place. Arrays of byte-strings are not swapped. The real and imaginary parts of a complex number are swapped individually.
If True, swap bytes in-place, default is False.
The byteswapped array. If inplace is True, this is a view to self.
Arrays of byte-strings are not swapped
but different representation in memory


method
Use an index array to construct a new array from a set of choices.
Refer to numpy.choose for full documentation.
See also
equivalent function


method
Return an array whose values are limited to [min, max]. One of max or min must be given.
Refer to numpy.clip for full documentation.
See also
equivalent function


method
Return selected slices of this array along given axis.
Refer to numpy.compress for full documentation.
See also
equivalent function


method
Complex-conjugate all elements.
Refer to numpy.conjugate for full documentation.
See also
equivalent function


method
Return the complex conjugate, element-wise.
Refer to numpy.conjugate for full documentation.
See also
equivalent function


method
Return a copy of the array.
Controls the memory layout of the copy. ‘C’ means C-order, ‘F’ means F-order, ‘A’ means ‘F’ if a is Fortran contiguous, ‘C’ otherwise. ‘K’ means match the layout of a as closely as possible. (Note that this function and numpy.copy are very similar but have different default values for their order= arguments, and this function always passes sub-classes through.)
See also
Similar function with different default behavior
This function is the preferred method for creating an array copy. The function numpy.copy is similar, but it defaults to using order ‘K’, and will not pass sub-classes through by default.


attribute
An object to simplify the interaction of the array with the ctypes module.
This attribute creates an object that makes it easier to use arrays when calling shared libraries with the ctypes module. The returned object has, among others, data, shape, and strides attributes (see Notes below) which themselves return ctypes objects that can be used as arguments to a shared library.
Possessing attributes data, shape, strides, etc.
See also
Below are the public attributes of this object which were documented in “Guide to NumPy” (we have omitted undocumented public attributes, as well as documented private attributes):
A pointer to the memory area of the array as a Python integer. This memory area may contain data that is not aligned, or not in correct byte-order. The memory area may not even be writeable. The array flags and data-type of this array should be respected when passing this attribute to arbitrary C-code to avoid trouble that can include Python crashing. User Beware! The value of this attribute is exactly the same as self._array_interface_['data'][0].
Note that unlike data_as, a reference will not be kept to the array: code like ctypes.c_void_p((a + b).ctypes.data) will result in a pointer to a deallocated array, and should be spelt (a + b).ctypes.data_as(ctypes.c_void_p)
(c_intp*self.ndim): A ctypes array of length self.ndim where the basetype is the C-integer corresponding to dtype('p') on this platform (see c_intp). This base-type could be ctypes.c_int, ctypes.c_long, or ctypes.c_longlong depending on the platform. The ctypes array contains the shape of the underlying array.
(c_intp*self.ndim): A ctypes array of length self.ndim where the basetype is the same as for the shape attribute. This ctypes array contains the strides information from the underlying array. This strides information is important for showing how many bytes must be jumped to get to the next element in the array.
Return the data pointer cast to a particular c-types object. For example, calling self._as_parameter_ is equivalent to self.data_as(ctypes.c_void_p). Perhaps you want to use the data as a pointer to a ctypes array of floating-point data: self.data_as(ctypes.POINTER(ctypes.c_double)).
The returned pointer will keep a reference to the array.
Return the shape tuple as an array of some other c-types type. For example: self.shape_as(ctypes.c_short).
Return the strides tuple as an array of some other c-types type. For example: self.strides_as(ctypes.c_longlong).
If the ctypes module is not available, then the ctypes attribute of array objects still returns something useful, but ctypes objects are not returned and errors may be raised instead. In particular, the object will still have the as_parameter attribute which will return an integer equal to the data attribute.


method
Return the cumulative product of the elements along the given axis.
Refer to numpy.cumprod for full documentation.
See also
equivalent function


method
Return the cumulative sum of the elements along the given axis.
Refer to numpy.cumsum for full documentation.
See also
equivalent function


attribute
Python buffer object pointing to the start of the array’s data.


method
Return specified diagonals. In NumPy 1.9 the returned array is a read-only view instead of a copy as in previous NumPy versions. In a future version the read-only restriction will be removed.
Refer to numpy.diagonal for full documentation.
See also
equivalent function


method
Dump a pickle of the array to the specified file. The array can be read back with pickle.load or numpy.load.
A string naming the dump file.
Changed in version 1.17.0: pathlib.Path objects are now accepted.


method
Returns the pickle of the array as a string. pickle.loads will convert the string back to an array.


method
Fill the array with a scalar value.
All elements of a will be assigned this value.


attribute
Information about the memory layout of the array.
The flags object can be accessed dictionary-like (as in a.flags['WRITEABLE']), or by using lowercased attribute names (as in a.flags.writeable). Short flag names are only supported in dictionary access.
Only the WRITEBACKIFCOPY, UPDATEIFCOPY, WRITEABLE, and ALIGNED flags can be changed by the user, via direct assignment to the attribute or dictionary entry, or by calling ndarray.setflags.
The array flags cannot be set arbitrarily:
Arrays can be both C-style and Fortran-style contiguous simultaneously. This is clear for 1-dimensional arrays, but can also be true for higher dimensional arrays.
Even for contiguous arrays a stride for a given dimension arr.strides[dim] may be arbitrary if arr.shape[dim] == 1 or the array has no elements. It does not generally hold that self.strides[-1] == self.itemsize for C-style contiguous arrays or self.strides[0] == self.itemsize for Fortran-style contiguous arrays is true.
The data is in a single, C-style contiguous segment.
The data is in a single, Fortran-style contiguous segment.
The array owns the memory it uses or borrows it from another object.
The data area can be written to. Setting this to False locks the data, making it read-only. A view (slice, etc.) inherits WRITEABLE from its base array at creation time, but a view of a writeable array may be subsequently locked while the base array remains writeable. (The opposite is not true, in that a view of a locked array may not be made writeable. However, currently, locking a base object does not lock any views that already reference it, so under that circumstance it is possible to alter the contents of a locked array via a previously created writeable view onto it.) Attempting to change a non-writeable array raises a RuntimeError exception.
The data and all elements are aligned appropriately for the hardware.
This array is a copy of some other array. The C-API function PyArray_ResolveWritebackIfCopy must be called before deallocating to the base array will be updated with the contents of this array.
(Deprecated, use WRITEBACKIFCOPY) This array is a copy of some other array. When this array is deallocated, the base array will be updated with the contents of this array.
F_CONTIGUOUS and not C_CONTIGUOUS.
F_CONTIGUOUS or C_CONTIGUOUS (one-segment test).
ALIGNED and WRITEABLE.
BEHAVED and C_CONTIGUOUS.
BEHAVED and F_CONTIGUOUS and not C_CONTIGUOUS.


attribute
A 1-D iterator over the array.
This is a numpy.flatiter instance, which acts similarly to, but is not a subclass of, Python’s built-in iterator object.
See also
Return a copy of the array collapsed into one dimension.
An assignment example:


method
Return a copy of the array collapsed into one dimension.
‘C’ means to flatten in row-major (C-style) order. ‘F’ means to flatten in column-major (Fortran- style) order. ‘A’ means to flatten in column-major order if a is Fortran contiguous in memory, row-major order otherwise. ‘K’ means to flatten a in the order the elements occur in memory. The default is ‘C’.
A copy of the input array, flattened to one dimension.
See also
Return a flattened array.
A 1-D flat iterator over the array.


method
Returns a field of the given array as a certain type.
A field is a view of the array data with a given data-type. The values in the view are determined by the given type and the offset into the current array in bytes. The offset needs to be such that the view dtype fits in the array dtype; for example an array of dtype complex128 has 16-byte elements. If taking a view with a 32-bit integer (4 bytes), the offset needs to be between 0 and 12 bytes.
The data type of the view. The dtype size of the view can not be larger than that of the array itself.
Number of bytes to skip before beginning the element view.
By choosing an offset of 8 bytes we can select the complex part of the array for our view:


method
Copy an element of an array to a standard Python scalar and return it.
A copy of the specified element of the array as a suitable Python scalar
When the data type of a is longdouble or clongdouble, item() returns a scalar array object because there is no available Python scalar that would not lose information. Void arrays return a buffer object for item(), unless fields are defined, in which case a tuple is returned.
item is very similar to a[args], except, instead of an array scalar, a standard Python scalar is returned. This can be useful for speeding up access to elements of the array and doing arithmetic on elements of the array using Python’s optimized math.


method
Insert scalar into an array (scalar is cast to array’s dtype, if possible)
There must be at least 1 argument, and define the last argument as item. Then, a.itemset(*args) is equivalent to but faster than a[args] = item. The item should be a scalar value and args must select a single item in the array a.
If one argument: a scalar, only used in case a is of size 1. If two arguments: the last argument is the value to be set and must be a scalar, the first argument specifies a single array element location. It is either an int or a tuple.
Compared to indexing syntax, itemset provides some speed increase for placing a scalar into a particular location in an ndarray, if you must do this. However, generally this is discouraged: among other problems, it complicates the appearance of the code. Also, when using itemset (and item) inside a loop, be sure to assign the methods to a local variable to avoid the attribute look-up at each loop iteration.


attribute
Length of one array element in bytes.


method
Return the maximum along a given axis.
Refer to numpy.amax for full documentation.
See also
equivalent function


method
Returns the average of the array elements along given axis.
Refer to numpy.mean for full documentation.
See also
equivalent function


method
Return the minimum along a given axis.
Refer to numpy.amin for full documentation.
See also
equivalent function


attribute
Total bytes consumed by the elements of the array.
Does not include memory consumed by non-element attributes of the array object.


method
Return the array with the same data viewed with a different byte order.
Equivalent to:
Changes are also made in all fields and sub-arrays of the array data type.
Byte order to force; a value from the byte order specifications below. new_order codes can be any of:
The default value (‘S’) results in swapping the current byte order.
New array object with the dtype reflecting given change to the byte order.


method
Return the indices of the elements that are non-zero.
Refer to numpy.nonzero for full documentation.
See also
equivalent function


method
Rearranges the elements in the array in such a way that the value of the element in kth position is in the position it would be in a sorted array. All elements smaller than the kth element are moved before this element and all equal or greater are moved behind it. The ordering of the elements in the two partitions is undefined.
New in version 1.8.0.
Element index to partition by. The kth element value will be in its final sorted position and all smaller elements will be moved before it and all equal or greater elements behind it. The order of all elements in the partitions is undefined. If provided with a sequence of kth it will partition all elements indexed by kth of them into their sorted position at once.
Deprecated since version 1.22.0: Passing booleans as index is deprecated.
Axis along which to sort. Default is -1, which means sort along the last axis.
Selection algorithm. Default is ‘introselect’.
When a is an array with fields defined, this argument specifies which fields to compare first, second, etc. A single field can be specified as a string, and not all fields need to be specified, but unspecified fields will still be used, in the order in which they come up in the dtype, to break ties.
See also
Return a parititioned copy of an array.
Indirect partition.
Full sort.
See np.partition for notes on the different algorithms.


method
Return the product of the array elements over the given axis
Refer to numpy.prod for full documentation.
See also
equivalent function


method
Peak to peak (maximum - minimum) value along a given axis.
Refer to numpy.ptp for full documentation.
See also
equivalent function


method
Set a.flat[n] = values[n] for all n in indices.
Refer to numpy.put for full documentation.
See also
equivalent function


method
Return a flattened array.
Refer to numpy.ravel for full documentation.
See also
equivalent function
a flat iterator on the array.


method
Repeat elements of an array.
Refer to numpy.repeat for full documentation.
See also
equivalent function


method
Returns an array containing the same data with a new shape.
Refer to numpy.reshape for full documentation.
See also
equivalent function
Unlike the free function numpy.reshape, this method on ndarray allows the elements of the shape parameter to be passed in as separate arguments. For example, a.reshape(10, 11) is equivalent to a.reshape((10, 11)).


method
Change shape and size of array in-place.
Shape of resized array.
If False, reference count will not be checked. Default is True.
If a does not own its own data or references or views to it exist, and the data memory must be changed. PyPy only: will always raise if the data memory must be changed, since there is no reliable way to determine if references or views to it exist.
If the order keyword argument is specified. This behaviour is a bug in NumPy.
See also
Return a new array with the specified shape.
This reallocates space for the data area if necessary.
Only contiguous arrays (data elements consecutive in memory) can be resized.
The purpose of the reference count check is to make sure you do not use this array as a buffer for another Python object and then reallocate the memory. However, reference counts can increase in other ways so if you are sure that you have not shared the memory for this array with another Python object, then you may safely set refcheck to False.
Shrinking an array: array is flattened (in the order that the data are stored in memory), resized, and reshaped:
Enlarging an array: as above, but missing entries are filled with zeros:
Referencing an array prevents resizing…
Unless refcheck is False:


method
Return a with each element rounded to the given number of decimals.
Refer to numpy.around for full documentation.
See also
equivalent function


method
Find indices where elements of v should be inserted in a to maintain order.
For full documentation, see numpy.searchsorted
See also
equivalent function


method
Put a value into a specified place in a field defined by a data-type.
Place val into a’s field defined by dtype and beginning offset bytes into the field.
Value to be placed in field.
Data-type of the field in which to place val.
The number of bytes into the field at which to place val.
See also


method
Set array flags WRITEABLE, ALIGNED, (WRITEBACKIFCOPY and UPDATEIFCOPY), respectively.
These Boolean-valued flags affect how numpy interprets the memory area used by a (see Notes below). The ALIGNED flag can only be set to True if the data is actually aligned according to the type. The WRITEBACKIFCOPY and (deprecated) UPDATEIFCOPY flags can never be set to True. The flag WRITEABLE can only be set to True if the array owns its own memory, or the ultimate owner of the memory exposes a writeable buffer interface, or is a string. (The exception for string is made so that unpickling can be done without copying memory.)
Describes whether or not a can be written to.
Describes whether or not a is aligned properly for its type.
Describes whether or not a is a copy of another “base” array.
Array flags provide information about how the memory area used for the array is to be interpreted. There are 7 Boolean flags in use, only four of which can be changed by the user: WRITEBACKIFCOPY, UPDATEIFCOPY, WRITEABLE, and ALIGNED.
WRITEABLE (W) the data area can be written to;
ALIGNED (A) the data and strides are aligned appropriately for the hardware (as determined by the compiler);
UPDATEIFCOPY (U) (deprecated), replaced by WRITEBACKIFCOPY;
WRITEBACKIFCOPY (X) this array is a copy of some other array (referenced by .base). When the C-API function PyArray_ResolveWritebackIfCopy is called, the base array will be updated with the contents of this array.
All flags can be accessed using the single (upper case) letter as well as the full name.


attribute
Number of elements in the array.
Equal to np.prod(a.shape), i.e., the product of the array’s dimensions.
a.size returns a standard arbitrary precision Python integer. This may not be the case with other methods of obtaining the same value (like the suggested np.prod(a.shape), which returns an instance of np.int_), and may be relevant if the value is used further in calculations that may overflow a fixed size integer type.


method
Sort an array in-place. Refer to numpy.sort for full documentation.
Axis along which to sort. Default is -1, which means sort along the last axis.
Sorting algorithm. The default is ‘quicksort’. Note that both ‘stable’ and ‘mergesort’ use timsort under the covers and, in general, the actual implementation will vary with datatype. The ‘mergesort’ option is retained for backwards compatibility.
Changed in version 1.15.0: The ‘stable’ option was added.
When a is an array with fields defined, this argument specifies which fields to compare first, second, etc. A single field can be specified as a string, and not all fields need be specified, but unspecified fields will still be used, in the order in which they come up in the dtype, to break ties.
See also
Return a sorted copy of an array.
Indirect sort.
Indirect stable sort on multiple keys.
Find elements in sorted array.
Partial sort.
See numpy.sort for notes on the different sorting algorithms.
Use the order keyword to specify a field to use when sorting a structured array:


method
Remove axes of length one from a.
Refer to numpy.squeeze for full documentation.
See also
equivalent function


method
Returns the standard deviation of the array elements along given axis.
Refer to numpy.std for full documentation.
See also
equivalent function