= Size-biased geometric distribution
{title2=$\widetilde Y\overset d=X_1+X_2-1,\quad X_i\sim\operatorname{Geom}(p)$}
For the positive-support <geometric distribution> with success probability $p$, <size bias> gives $\mathbb P(\widetilde Y=k)=kp^2(1-p)^{k-1}$. 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 $p^2s/(1-(1-p)s)^2$, also obtained by the <size-biased probability generating function> identity.
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