Question

How to derie likelihood and loglikelihood function of truncated poisson distribution?

Answer 1

I am trying to construct a likelihood function for a truncated poisson distribution, the formula for my probability mass function is:

fX(x)=λx(eλ−1)(x!)fX(x)=λx(eλ−1)(x!)

I know that in order to construct a likelihood function I need to take the product of each instance of xixi from 1 to n, i.e.

∏i=1nλx(eλ−1)(x!)∏i=1nλx(eλ−1)(x!)

My only problem is that I can't work out how to handle the x!x! term in my working. Thus far, I have:

L(λ)=λ∑(xi)(eλ−1)n(∏x!)L(λ)=λ∑(xi)(eλ−1)n(∏x!)

This doesn't feel complete though as I feel the product over factorials term can be somehow simplified using the gamma function