# Pseudo-random number sampling

Pseudo-random number sampling or non-uniform pseudo-random variate generation is the numerical practice of generating pseudo-random numbers that are distributed according to a given probability distribution.

Methods of sampling a non-uniform distribution are typically based on the availability of a pseudo-random number generator producing numbers X that are uniformly distributed. Computational algorithms are then used to manipulate a single random variate, X, or often several such variates, into a new random variate Y such that these values have the required distribution.

Historically, basic methods of pseudo-random number sampling were developed for Monte-Carlo simulations in the Manhattan project; they were first published by John von Neumann in the early 1950s.

## Finite discrete distributions

For a discrete probability distribution with a finite number n of indices at which the probability mass function f takes non-zero values, the basic sampling algorithm is straightforward. The interval [0, 1) is divided in n intervals [0, f(1)), [f(1), f(1) + f(2)), ... The width of interval i equals the probability f(i). One draws a uniformly distributed pseudo-random number X, and searches for the index i of the corresponding interval. The so determined i will have the distribution f(i).

Formalizing this idea becomes easier by using the cumulative distribution function

$F(i)=\sum _{j=1}^{i}f(j).$ It is convenient to set F(0) = 0. The n intervals are then simply [F(0), F(1)), [F(1), F(2)), ..., [F(n  1), F(n)). The main computational task is then to determine i for which F(i  1)  X < F(i).

This can be done by different algorithms:

## Continuous distributions

Generic methods for generating independent samples:

Generic methods for generating correlated samples (often necessary for unusually-shaped or high-dimensional distributions):

For generating a normal distribution:

For generating a Poisson distribution:

## Software libraries

GNU Scientific Library has a section entitled "Random Number Distributions" with routines for sampling under more than twenty different distributions.

## Footnotes

1. Von Neumann, John (1951). "Various Techniques Used in Connection with Random Digits" (PDF). Journal of Research of the National Bureau of Standards, Applied Mathematics Series. 3: 36–38. Any one who considers arithmetical methods of producing random digits is of course, in a state of sin. Also online is a low-quality scan of the original publication.
2. Ripley (1987)
3. Fishman (1996)
4. Fishman (1996)

## Literature

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