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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?

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This is very hard to read because: (1) You seem to have left out the squares when typing your equation and (2) There is a great deal of discussion of your emotional state, and little of the problem. How about just "I am trying to compute the variance of ... This means I need to evaluate the sum ... Does anyone know how to do this?" –  David Speyer Jan 27 '10 at 20:47
I've cleaned up the equations a bit. –  vilvarin Jan 27 '10 at 20:49

3 Answers 3

up vote 2 down vote accepted

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.

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the previous answer has vanished. To tell the truth I haven't understood it completly :(

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thanks, David. The original problem I had is to compute the variance Y/(X+1) where Y bernoulli distributed with parameter p , and X is poisson distributed. But I don't think it will change anything. So I will just leave the sum (Vilvarin)

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