Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability (resit) / 8 / a

The probability generating function of a nonnegative integer-valued random variable isusing $s^0=1$ also at $s=0$. It is always defined for $0\le s\le 1$, and indeed the power series converges absolutely for all real or complex $s$ with $|s|\le 1$, becauseIt may be defined beyond the unit disk where this power series converges absolutely; such a larger domain depends on the distribution. In particular $G_X(0)=\mathbb{P}(X=0)$ and $G_X(1)=1$.

Let $S_n={\sum }_{i=1}^n{X}_i$. By independence, the expectation of a product of independent random variables factors, and the identical distributions giveThis is valid for $|s|\le 1$ and wherever the required absolute expectations exist. It is the probability generating function of an independent sum.

For $0\le s\le 1$, condition on $N=n$. Its independence from the $X_i$ preserves their independent common distribution, so part (ii) givesFor $n=0$, this is $1$, agreeing with $Y=0$. By the law of total expectation,This proves the probability generating function of an independent random sum. For nonnegative $s$ outside $[0,1]$, the same argument is valid whenever the right side is finite, by Tonelli's theorem for nonnegative sums. For signed or complex $s$, a sufficient absolute-convergence condition is $G_N(G_X(|s|))<\infty$; it holds automatically on $|s|\le 1$. Outside this domain, mere evaluation of an analytically extended expression is not a substitute for existence of the probability generating function.