I was wondering if it was possible to get the EV and Variance of the sum of claims($S$) using a compounded distribution, given that: $$ N \sim Po(\theta) \\ \theta \sim U(0,0.1) \\ X \sim Exp(\dfrac{1}{10}) \\ S = \sum_{i=1}^{N} X_{i} $$ I have already found $E(s)$ and $Var(S)$ using an easy approach (conditional expected values and variances) but I was wondering if it was possible to use a Compounded Poisson or Compounded Negative Binomial distribution for $S$.

I have read in $\textit{Actuarial Mathematics}$, by N. L. Bowers, that this was a possible alternative and, for this case, since $E(N)<Var(N)$, $N$ would have a Negative Binomial Distribution, hence $S$ would adopt a Compounded NB form.

I would appreciate any help in understanding if the approach is correct and how to apply it in order to get a numeric representation of the distribution.

Thanks in advance!


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