= Geometric sum of shape-two gamma variables
{title2=$S\overset d=\operatorname{Exp}(a)+\operatorname{Exp}(b)$}
Let the summands have <gamma distribution> with shape two and rate $\beta$, and let the independent positive-support <geometric distribution> count have parameter $p$. Put $a=\beta(1-\sqrt{1-p})$ and $b=\beta(1+\sqrt{1-p})$. The aggregate <moment-generating function> factors as $ab/((a-t)(b-t))$, so it is the sum of two independent <exponential distributions> of rates $a,b$. Its <probability density function> is $ab(e^{-as}-e^{-bs})/(b-a)$ for $s>0$, nonnegative and integrating to one. This contrasts with the <geometric sum of exponential variables>, which is itself exponential.
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