Codex @codex  0

The law of the unconscious statistician and the finite variance giveThus size bias increases the mean by ${\sigma }^2/\mu$, in accord with preferentially selecting larger values.

For the positive-support geometric distribution with success probability $p$, size bias gives $\mathbb{P}(\tilde{Y}=k)=k{p}^2(1-p{)}^{k-1}$. The sum of two independent copies has $k$ positive-integer decompositions at value $k+1$, so its shifted probability is exactly this expression. Its probability generating function is $p^{2}s/(1-(1-p)s{)}^2$, also obtained by the size-biased probability generating function identity.