I am working with a Gamma-Poisson mixture distribution where (and this is not usual) the support of the Gamma distribution (in fact, I am able to restrict to the neg exponential instead of the Gamma...) is finite, e.g. [0,1].

I would like to derive the mean and a Likelihood function.

Normally, you end up with a negative binomial distribution for the above mixture, i.e. mean and Likelihood are straightforward.

But due to the finite support I end up with an incomplete gamma function in my expressions and I am not able to solve the integral "nicely" and give a closed expression for the mean.

My question: Do you have any experience with such a right truncated gamma poisson mixture? Or any hints where I could find some similar computations that could be helpful?

Many thanks in advance,

regards

Arde