The Poisson random variable is discrete, having a PMF fx(xi) = P (X = xi) given in Eq. (2.53). Dagpunar (1988) presented a simple algorithm and used the CDF-inverse method based on Eq. (6.7). When generating Poisson random variates, care should be taken so that e~v is not smaller than the machine’s smallest positive real value. This could occur especially when the Poisson
parameter v is large. An algorithm for generating Poisson random variates is as follows:
1. Generate u ~ U(0,1) and initialize x = 0 and y = e~v
2. If y < u, go to step 3. Otherwise, x is the Poisson random variate sought.
3. Let u = u — y, x = x + 1, and update y = vy/x. Then go to step 2.
This algorithm is efficient when v < 20. For a large v, the Poisson distribution can be approximated by a normal distribution with a mean v — 0.5 and standard deviation of>. Then a Poisson random variate is set to the round-off normal random variate from N(v — 0.5^/v).
Other algorithms have been developed for generating Poisson random variates. Rubinstein (1981) used the fact that the interarrival time between events for a Poisson process has an exponential distribution with parameter 1/v. Atkinson (1979) applied the AR method using a logistic distribution as the enveloping PDF.
