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

For a nonnegative integer-valued random variable, its probability generating function is $G_X(s)=\mathbb{E}[s^X]={\sum }_{k\ge 0}\mathbb{P}(X=k)s^k$, at least for $|s|\le 1$. Insert the Poisson distribution masses and sum the exponential series:

Independence gives the probability generating function of the sum as the product:This is the probability generating function of the Poisson distribution with parameter $\mu +\lambda$. Uniqueness of the coefficients of a power series therefore gives $X+Y\sim \mathrm{Poisson}(\mu +\lambda )$. The product rule for these generating functions requires the stated independence.

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.

The conditional mean of a Poisson distribution equals its parameter, so $\mathbb{E}[Z\mid X]=X$. The law of total expectation givesAs a check, differentiating the probability generating function in part (iii) at one also gives $G_Z^{\prime }(1)=\mu$.