Let $X$ be a random variable taking non-negative integer values and let $Y$ be a random variable taking real values.

(a) Define the probability-generating function $G_{X}(s)$. Calculate it explicitly for a Poisson random variable with mean $\lambda>0$.

(b) Define the moment-generating function $M_{Y}(t)$. Calculate it explicitly for a normal random variable $\mathrm{N}(0,1)$.

(c) By considering a random sum of independent copies of $Y$, prove that, for general $X$ and $Y, G_{X}\left(M_{Y}(t)\right)$ is the moment-generating function of some random variable.