Some properties of the 'hermite' distribution.

SUMMARY The Hermite distribution is the generalized Poisson distribution whose probability generating function (P.G.F.) is exp [a1(s - 1) + a2(s2 - 1)]. The probabilities (and factorial moments) can be conveniently expressed in terms of modified Hermite polynomials (hence the proposed name). Writing these in confluent hypergeometric form leads to two quite different representations of any particular probability-one a finite series, the other an infinite series. The cumulants and moments are given. Necessary conditions on the parameters and their maximum-likelihood estimation are discussed. It is shown, with examples, that the Hermite distribution is a special case of the Poisson Binomial distribution (n = 2) and may be regarded as either the distribution of the sum of two correlated Poisson variables or the distribution of the sum of an ordinary Poisson variable and an (independent) Poisson 'doublet' variable (i.e. the occurrence of pairs of events is distributed as a Poisson). Lastly its use as a penultimate limiting form of distributions with P.G.F.