Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 2 / 3F (Probability)

For a nonnegative integer-valued random variable, the probability generating function isinitially for $|s|\le 1$, with $0^0=1$. For the Poisson distribution, insert its probability mass function and sum the exponential series:Thus

The moment-generating function is $M_Y(t)=\mathbb{E}[e^{tY}]$ for real $t$ where this expectation is finite; a finite neighborhood of zero is usually required when speaking of an existing moment-generating function. For a standard normal distribution, completing the square givesThe translated Gaussian density integrates to one, so

Construct independent copies $Y_1,Y_2,\dots$ of $Y$, also independent of $X$, and define the random sum $Z={\sum }_{j=1}^X{Y}_j$, taking the empty sum to be zero. Independence gives, conditional on $X=n$,The law of total expectation gives the transform composition for an independent random sumThe summands are nonnegative, so this equality is valid also as an extended-valued expectation. Where the right-hand side is finite, it is the ordinary moment-generating function of the constructed random sum. The construction does not assert a finite moment-generating function near zero for arbitrary $X,Y$: heavy tails can prevent this. For example a nonnegative $Y$ with no positive exponential moments and $X\equiv 1$ already shows the necessary qualification.