Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 41 / 1 / i

Let $\Lambda ={\sum }_{i=1}^m{\lambda }_i$. For $\Lambda >0$, the resulting compound Poisson distribution hasThus the merged claim size probability distribution is a rate-weighted mixture distribution, not an equally weighted mixture distribution unless the rates agree.

To prove this without any moment-generating function assumption, let ${\chi }_i(t)$ be the characteristic function of a claim of type $i$. Conditioning on its Poisson distribution count givesIndependence of the aggregate risks therefore givesThe weighted characteristic function inside is precisely that of the displayed mixture distribution. This is the transform of a sum of independent claims with independent count $\mathrm{Pois}(\Lambda )$, proving the claim by uniqueness of characteristic functions. This is Poisson superposition of insurance portfolios. If $\Lambda =0$, all risks make no claims almost surely and $S=0$; the severity law is then irrelevant.