This question came up during my research. In detail, let $X$ be Poisson distribution, i.e. $X\sim Pois(\lambda)$. I am interested in the expected value

$$ \mathbb{E}\left[\log\left(\frac{\Gamma(X+c)}{\Gamma(X+d)}\right)\right],\qquad (1)$$

where $c,d>0$.

My idea to calculate the expected value was to calculate first the expected value $\mathbb{E}\left[\log\left(\Gamma(X+c)\right)\right]$ based on some series expansion of the log Gamma function. However, this resulted in being stuck at the expected value of $\mathbb{E}\left[\log\left(X+c\right)\right]$.

Does anybody have some advice on this?

Edit based on comments below:

  • $c$ and $d$ can but do not have to be integers.
  • The Gamma function is defined by $\Gamma(z) = \int_0^\infty x^{z-1} e^{-x}\,dx$.
  • Delta method: It was suggested to consider the Delta method. I think it does not really apply here as I do not have a sequence of random variables. That being said, I compared (1) with $ \log\left(\frac{\Gamma(\lambda+c)}{\Gamma(\lambda+d)}\right)$ and the difference is generally too big to consider this approximation. Here is the R code:

    lambda <- 3
    c <- 1.5
    d <- 3.2
    x <- seq(0, 2000)
    sum((lgamma(x + c) - lgamma(x + d)) * dpois(x = x, lambda = lambda))
    lgamma(lambda + c) - lgamma(lambda + d)
  • $\begingroup$ Can you define $\Gamma$ also? I understand it is Gamma function, but just for the completeness of the question. $\endgroup$
    – Maxtron
    Sep 12, 2018 at 19:20
  • $\begingroup$ Delta method ... ?? Don't know if that will be good enough for your application. $\endgroup$
    – Ben Bolker
    Sep 12, 2018 at 19:33
  • 6
    $\begingroup$ $c$ and $d$ wouldn't, by great good fortune, happen to be integers, would they? $\endgroup$
    – jbowman
    Sep 12, 2018 at 19:37
  • $\begingroup$ Thank you, everybody. I edited the original question to address your comments. $\endgroup$
    – Nussig
    Sep 13, 2018 at 6:58
  • $\begingroup$ I'm still wondering for a non-Taylor answer even if $c$ and $d$ were integers. $\endgroup$
    – gunes
    Sep 13, 2018 at 6:59


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