Non-uniform random variate generation

Generating pseudo-random numbers that follow a probability distribution From Wikipedia, the free encyclopedia

Non-uniform random variate generation or pseudo-random number sampling is the numerical practice of generating pseudo-random numbers (PRN) that follow a given probability distribution. Methods are typically based on the availability of a uniformly distributed PRN generator. 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. The first methods were developed for Monte-Carlo simulations in the Manhattan project,[citation needed] published by John von Neumann in the early 1950s.[1]

Finite discrete distributions

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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

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

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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.[5]

See also

Footnotes

Literature

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