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