What is the variance of $1/(X+1)$ where $X$ is Poisson-distributed with parameter $\lambda$! The series for the second moment is horrible!


$E({1\over (X+1)^2})=\sum\_{k=1}^{\infty}\frac{1}{k^{2}}\frac{\lambda^{k}e^{-\lambda}}{k!}$


Is there an easy way to do it?



variance of $1/(X+1)$ where $X$ is Poisson-distributed with parameter $\lambda$ 
Sorry, I gave a moronic answer before. Let me try to give a better one.


There should be no expression for $f(\lambda) := \sum\_{k \geq 1} \lambda^k/(k^2 k!)$ in elementary functions. If there were, then $g(\lambda) = \lambda f'(\lambda) = \sum\_{k \geq 1} \lambda^{k}/(k \cdot k!)$ would also be elementary. But $g(\lambda)=\int\_0^{\lambda} \frac{e^t-1}{t} dt$ and $e^t/t$ is a standard example of a function without an elementary antiderivative.


