Let $X = \sum_{i=1}^N X_i$ where all $X_i$'s are iid and $N$ and $X_i$ are independent where $N$ is a positive-valued integer r.v.
Then the PGF of $X$ is
$$G_{X}(s) = \mathbb{E}(s^X)
$$
They simplified this down to:
$$G_{X}(s) = \mathbb{E}(\prod_{i=1}^N \mathbb{E}(s^{X_i})) \\
= \mathbb{E}(G_{X_i}(s)^N)\\
= G_{N}(G_{X_i}(s)) \\
= \sum_{n=0}^\infty (G_{X_i}(s))^n \mathbb{P}(N=n)
$$
My question is: why do they use the probability density function of $N$ to compute the expectation in the end? So how did they go from the third last equality to the second last?
Because it seems unclear whether to use $X_i$ or $N$, since $X_i$ is also a random variable here. Can't I do this:
$$
G_{X}(s) = \mathbb{E}(G_{X_i}(s)^N) = \mathbb{E}((\exp(N)))^{\log(G_{X_i}(s))})
$$
and then use the PDF of $X_i$?
I think my question boils down to: how do I know what PDF to use?