Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 9 / a / iii / Solution

Condition first on $X$. A Poisson distribution with parameter $k$ has conditional probability generating function $e^{k(s-1)}$, including the degenerate zero case when $k=0$. The law of total expectation givesFor $0\le s\le 1$ the inner argument is in $(0,1]$, so this use of the generating function is justified directly. The result is a compound Poisson distribution with a Poisson distribution of unit parameter for each jump, also called a Neyman Type A distribution in this case. It is generally not an ordinary Poisson distribution.