variance of $1/(X+1)$ where $X$ is Poisson-distributed with parameter $\lambda$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T21:34:44Z http://mathoverflow.net/feeds/question/13181 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13181/variance-of-1-x1-where-x-is-poisson-distributed-with-parameter-lambda variance of $1/(X+1)$ where $X$ is Poisson-distributed with parameter $\lambda$ vilvarin 2010-01-27T20:31:38Z 2010-01-27T21:27:52Z <p>What is the variance of $1/(X+1)$ where $X$ is Poisson-distributed with parameter $\lambda$! The series for the second moment is horrible!</p> <p>$E({1\over (X+1)^2})=\sum_{k=1}^{\infty}\frac{1}{k^{2}}\frac{\lambda^{k}e^{-\lambda}}{k!}$</p> <p>Is there an easy way to do it?</p> http://mathoverflow.net/questions/13181/variance-of-1-x1-where-x-is-poisson-distributed-with-parameter-lambda/13189#13189 Answer by David Speyer for variance of $1/(X+1)$ where $X$ is Poisson-distributed with parameter $\lambda$ David Speyer 2010-01-27T21:21:54Z 2010-01-27T21:21:54Z <p>Sorry, I gave a moronic answer before. Let me try to give a better one.</p> <p>There should be no expression for <code>$f(\lambda) := \sum_{k \geq 1} \lambda^k/(k^2 k!)$</code> in elementary functions. If there were, then <code>$g(\lambda) = \lambda f'(\lambda) = \sum_{k \geq 1} \lambda^{k}/(k \cdot k!)$</code> would also be elementary. But <code>$g(\lambda)=\int_0^{\lambda} \frac{e^t-1}{t} dt$</code> and $e^t/t$ is a standard example of a function without an elementary antiderivative.</p> http://mathoverflow.net/questions/13181/variance-of-1-x1-where-x-is-poisson-distributed-with-parameter-lambda/13190#13190 Answer by vilvarin for variance of $1/(X+1)$ where $X$ is Poisson-distributed with parameter $\lambda$ vilvarin 2010-01-27T21:22:25Z 2010-01-27T21:22:25Z <p>the previous answer has vanished. To tell the truth I haven't understood it completly :(</p> http://mathoverflow.net/questions/13181/variance-of-1-x1-where-x-is-poisson-distributed-with-parameter-lambda/13192#13192 Answer by vilvarin for variance of $1/(X+1)$ where $X$ is Poisson-distributed with parameter $\lambda$ vilvarin 2010-01-27T21:27:52Z 2010-01-27T21:27:52Z <p>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)</p>