Generating Poisson-distributed Random Variables
A simple algorithm to generate random Poisson-distributed numbers (pseudo-random number sampling) has been given by Knuth (see References below):
algorithm poisson random number (Knuth): init: Let L ← e−λ, k ← 0 and p ← 1. do: k ← k + 1. Generate uniform random number u in and let p ← p × u. while p > L. return k − 1.While simple, the complexity is linear in λ. There are many other algorithms to overcome this. Some are given in Ahrens & Dieter, see References below. Also, for large values of λ, there may be numerical stability issues because of the term e−λ. One solution for large values of λ is Rejection sampling, another is to use a Gaussian approximation to the Poisson.
Inverse transform sampling is simple and efficient for small values of λ, and requires only one uniform random number u per sample. Cumulative probabilities are examined in turn until one exceeds u.
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