Bayesian Statistics: An Introduction, 4th Edition

A.12 Poisson distribution

X has a Poisson distribution of mean λ, denoted

if it has a discrete distribution with density

The mean and variance are

Because

we see that p(X+1)> p(X) if and only if

and hence that a mode occurs at

the square brackets denoting ‘integer part of’, and this mode is unique unless λ is an integer.

Integrating by parts shows that the distribution function is expressible in terms of the incomplete gamma function, namely,

see Kendall, Stewart and Ord (1987, Section 5.9).

The Poisson distribution often occurs as the limit of the binomial as