Shifted-exponential Poisson mixture
= Shifted-exponential Poisson mixture
{title2=$G(z)=e^{\alpha(z-1)}\beta/(\beta+1-z)$}
When the random intensity is $\alpha+Y$ with $Y$ having an <exponential distribution> of rate $\beta$, the mixed count is the sum of independent <Poisson distribution> and support-zero <geometric distribution> counts. The Poisson parameter is $\alpha$ and the geometric success probability is $\beta/(\beta+1)$. At $\alpha=0$ the law is geometric.