PFG of Y~Bin(n,p)

Question

Let $X_1, X_2, \ldots , X_n \sim \operatorname{Ber}(p)$ be iid Bernoulli random variables.

(a) Determine the probability generating function of $Y \sim \operatorname{Bin}(n, p)$.

I understand that for $X \sim \operatorname{Bin} (n,p)$, the general P.G.F is $G_X (s) = (q + ps)^n$ , $(q=1-p)$. But I don't understand how to start off with it in terms of Y. Please help me get started.

Answer 1

The PGF is $G_Y(s) = \mathsf E(s^Y)$

Now $Y$ is the series of of independent and identically distributed Bernoulli variables. $$Y=\sum\limits_{k=1}^n X_k$$

The exponential of a series is a product exponentials...

$$s^Y= \prod\limits_{k=1}^n s^{X_k}$$

Then, because of independence and identical distribution: $$\mathsf E(\prod_{k=1}^n s^{X_k}) ~=~ \mathsf E(s^{X_1})^n$$

Then what is $\mathsf E(s^{X_1})$ for $X_1\sim\mathcal {Bin}(1,p)$ ?