Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 2 / 10F / a / Solution

Conditional on $N=n$, the $n$ type indicators are independent random variables with the Bernoulli distribution of parameter $p$. Thus $F\mid N=n$ has the binomial distribution and its probability generating function is $(1-p+ps{)}^n$. The general rule used here is that for a nonnegative integer-valued random variable $X$, $G_X(s)=\mathbb{E}[s^X]$; for a sum of independent such variables, their probability generating functions multiply, since the expected value of the product $s^{X_1}\cdots s^{X_n}$ factors. The law of total expectation then averages the conditional result:This calculation uses the given independent labeling of objects, including conditional on their number.