Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Mixed Poisson Models for Claim Numbers 27
Shaked’s Two Crossings Theorem tells us (i) that the mixed Poisson distribution has an
excess of zeros compared to the Poisson distribution with the same mean and (ii) that the
mixed Poisson distribution has a thicker right tail than the Poisson distribution with the
same mean.
Probability Generating Function
The probability generating function of Poisson mixtures is closely related to the moment
generating function of the underlying random effect. Moment generating functions are a
widely used tool in many statistics texts, and also in actuarial mathematics. They serve
to prove statements about convolutions of distributions, and also about limits. Recall that
the moment generating function of the nonnegative random variable X, denoted as M X ,is
given by
M X t = EexptX t > 0
It is interesting to mention that M X characterizes the probability distribution of X, i.e. the
information contained in F X and M X is equivalent.
If there exists h>0 such that M X t exists and is finite for 0