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\title{Nested Poisson}
\author{Christopher D'Arcy}
\date{}
\begin{document}
\maketitle
\section{Nested Poisson PMF}
The PMF of a Poisson random variable (denoted $Po(\lambda)$) with constant rate $\lambda$ is as follows:
\begin{eqnarray}
P(X=k|X\sim Po(\lambda ))=\frac{e^{-\lambda}\lambda^k}{k!}
\end{eqnarray}
We can see that the PMF of a Poisson random variable generated from a rate proportional by a constant $\mu$ to a Poisson random variable with a rate $\lambda$ is equal to the following due to the law of total probability:
\begin{eqnarray}
P(X=k|X\sim Po(\mu Po(\lambda )))=\sum^{\infty}_{\tau=0}\frac{e^{-\mu \tau}(\mu \tau)^k}{k!}\cdot \frac{e^{-\lambda}\lambda^\tau}{\tau !}
\end{eqnarray}
Setting the constant $e^{-\mu}\lambda=\alpha$ we see:
\begin{eqnarray}
\sum^{\infty}_{\tau=0}\frac{e^{-\mu \tau}(\mu \tau)^k}{k!}\cdot \frac{e^{-\lambda}\lambda^\tau}{\tau !}=\sum^{\infty}_{\tau=0}\frac{e^{-\lambda}(\mu \tau)^k}{k!}\cdot \frac{\alpha^{\tau}}{\tau !}=\frac{e^{-\lambda}\mu^{k}}{k!}\sum^{\infty}_{\tau=0}\frac{\tau^k\alpha^{\tau}}{\tau !}
\end{eqnarray}
As we know that $e^z=\sum^{\infty}_{n=0}\frac{z^{n}}{n!}$ we can then show Touchards $k^{th}$ polynomial is of use in reducing the equation:
\begin{eqnarray}
\frac{e^{-\lambda}\mu^{k}}{k!}\sum^{\infty}_{\tau=0}\frac{\tau^k\alpha^{\tau}}{\tau !}=\frac{e^{\alpha-\lambda}\mu^{k}}{k!}T_{k}(\alpha )
\end{eqnarray}
Where $T_k(\alpha )$ is Touchard's $k^{th}$ polynomial. Thus:
\begin{eqnarray}
P(X=k|X\sim Po(\mu Po(\lambda )))=\frac{e^{\alpha-\lambda}\mu^{k}}{k!}T_{k}(\alpha )
\end{eqnarray}
\end{document}
