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I found the expression for the expected value of the falling factorial of a Poisson distribution ($\lambda^n$) from - Is there a similar expression for the expected values of the rising factorial and plain simple factorial? I am actually after the expected value of the logarithms of these quantities: $$\sum_{u=1}^{X}log(u)$$ and $$\sum_{u=X}^{X+c}log(u)$$ where $X \to Poisson(\lambda)$.

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If you just need an approximation, a normal approximation may work well. – Douglas Zare Apr 2 '13 at 8:06
the expectation value of the plain simple factorial is obviously just $e^{−\lambda}(1−\lambda)^{−1}$, if $\lambda<1$ and divergent otherwise; the expectation value of the log of the factorial has no closed form expression. – Carlo Beenakker Apr 2 '13 at 21:36

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