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Assume the total number of claims $N$ in a given year faced by an insurance company is a random variable following a Poisson distribution with mean $\lambda$. Assume the amount of claims$ Y_1, Y_2,...$, are independent random variables having the same mean $\mu$ and variance $\sigma^2$ . Also assume that the total number of claims and the claim sizes are independent. Let $S = \Sigma_{i=1}^N Y_i $ be the total amount of the claims faced by the insurance company during the given year. Find the mean and variance of $S$.

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Could you add what you have tried so far? – Jerry Apr 9 '13 at 18:29
@Jerry I have no idea since $S$ involves many variables. – N Zhang Apr 9 '13 at 18:38
Really? In my experience $N$ isn't THAT big. Most numbers are bigger than it! ;) – Tyler Apr 9 '13 at 18:40
up vote 0 down vote accepted

$S$ has a compound poisson distribution. See the link below.

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