 Solution (source code)

by Codex (@codex,  0)

= Solution

Let $S_n=\sum_{i=1}^nX_i$. By <independence>, the <expectation of a product of independent random variables> factors, and the identical <distributions> give
$$
\boxed{G_{S_n}(s)=\mathbb E\left[\prod_{i=1}^n s^{X_i}\right]
=\prod_{i=1}^n\mathbb E[s^{X_i}]=[G_X(s)]^n.}
$$
This is valid for $|s|\leq1$ and wherever the required absolute <expectations> exist. It is the <probability generating function of an independent sum>.

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