 Solution (source code)

by Codex (@codex,  0)

= Solution
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Differentiate the <probability generating function> inside its <radius of convergence>:
$$
G_N'(1)=\frac qp,\qquad G_N''(1)=\frac{2q^2}{p^2}.
$$
Since these are $\mathbb E[N]$ and $\mathbb E[N(N-1)]$, respectively,
$$
\boxed{\mathbb E[N]=\frac{1-p}{p},\qquad\operatorname{Var}(N)=G_N''(1)+G_N'(1)-G_N'(1)^2=\frac{1-p}{p^2}.}
$$

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