Claims per policyholder follows a Poisson dist. but mean varies according to a Gamma distribution

Question

Can anyone help me on this?

The number of claims per policyholder in a portfolio follows a Poisson distribution with a mean number $Q$. Suppose that the mean number $Q$ varies over the policyholder population according to a Gamma distribution. Derive a formula for the probability of a randomly chosen policyholder making $r$ claims and why this may be more appropriate in many applications than the Poisson model?

Can anyone give any tip on how to do this and where can I find information that can help me?

Thanks

Answer 1

Comment continued. Following both my Comment and the one of @spaceisdarkgreen, here is a simulation based on the specific case $Q \sim \mathsf{Gamma}(\text{shape}= \alpha = 3,\, \text{rate} = \beta = 1/5).$ For an actuarial application, notice that the standard deviation of this distribution is greater than the SD of a Poisson distribution with mean 15. (For a general mathematical formula, you can adjust the notation in Wikipedia to match the notation in your Question.)

set.seed(310)  # retain this statement to get exactly same simulation; otherwise omit
m = 10^6; r = numeric(m)
al = 3;  be = 1/5
for(i in 1:m) {
  q = rgamma(1, al, be)
  r[i] = rpois(1, q) }
mean(r);  sd(r)
## 14.98479
## 9.47612