I'm reading Dr Gleb Gribakin's notes on Probability and Distribution Theory and hit an problem in Chapter 3. Probability generating functions:
Let $N$, $X_1$, $X_2$, $\dots$ be independent count r.v.s. If the $\{X_i\}$ are identically distributed, each with PGF (probability generating function) $G_X$, then $S_N := X_1 + \cdots + X_N$ has PGF $$G_{S_N} = G_N (G_X(s))$$
And Corollaries 2: $$Var(S_N) = E(N) Var(X) + Var(N) [E(X)]^2$$
I'm a bit lost, how to prove Corollaries 2?
So far I've tried:
First, for any count r.v. $X$ and its PGF $G_X(s)$, we have: $$ G'_X(1) = E(x)$$ and $$ Var(X) = G''_X(1) - [G'_X(1)]^2 + G'_X(1)$$
Then, denote $u = G_X(s)$, we have $$\frac{d}{ds} G_{S_N}(s) = \frac{d}{ds}[G_N(G_X(s))] = \frac{d G_N(u)}{du} \cdot \frac{du}{ds}$$ , so $$E(S_N) = \frac{d}{ds} G_{S_N}(1) = G'_N(1) \cdot G'_X(1) = E(N) E(X)$$
Derive once more, we have
$$\frac{d^2}{ds^2} G_{S_N}(1) = \left[ \frac{d^2 G_N(1)}{du^2} \right] \cdot \left( \frac{dG_N(1)}{ds} \right)^2 + \frac{d G_N(1)}{du} \cdot \frac{d^2 G_X(1)}{ds^2} $$ $$= [Var(N) + E^2(N) - E(N)]\cdot E^2(X) + E(N) \cdot [Var(X) + E^2(X) - E(X)] $$ $$ = Var(N) E^2(X) + Var(X) E(N) + E^2(N) E^2(X) - E(N) E(X)$$
But it does not tally with the Corollaries 2.
Where did I miss?
You are almost there. You showed that $$ G''_{S_N}(1) = \operatorname{Var}[N] \mathbb{E}[X]^2 + \operatorname{Var}[X] \mathbb{E}[N] + \mathbb{E}[N]^2 \mathbb{E}[X]^2 - \mathbb{E}[N] \mathbb{E}[X] \tag{1} $$ but you are not quite done. This is not yet the variance of $S_N$, which is $$ \operatorname{Var}[S_N] = G''_{S_N}(1) - \mathbb{E}[S_N]^2 + \mathbb{E}[S_N] \tag{2} $$
Since you have shown that $\mathbb{E}[S_N] = \mathbb{E}[N]\mathbb{E}[X]$, combining (1) and (2) gives $$ \operatorname{Var}[S_N] = \operatorname{Var}[N] \mathbb{E}[X]^2 + \operatorname{Var}[X] \mathbb{E}[N]\tag{3} $$ as desired.
For context, the equality $$\mathbb{E}[S_N] = \mathbb{E}[N]\mathbb{E}[X]$$ is an instance of Wald's identity.