I was just working through this question.
A compound Poisson risk model is used to model the total claims S experienced by an insurance company over one year, of the form:
$S = X_1 + ... + X_n$
where $ X_i$ represents the size of claim $i$ and $N$ is the total number of claims, following a Poisson distribution. Each claim size $X_i$ is distributed according to $X_i$ ~ $Gamma(α,λ) $
In the 5 preceding years the average number of claims per year has been 14. The average claim size has been 500 and the standard deviation of claim sizes has been 150.
I am supposed to estimate:
a) $α$ and $λ $ for the Gamma distribution
b) The variance and mean for the total number of claims
My attempt at a solution, I am not sure if correct.
a) I used the method of moments.The first moment (i.e. mean) is given by $E(X)= α/λ $, and the second moment (i.e. variance) is given by $ Var(X) = \alpha/\lambda^2 $. Setting E(X) = 500, and Var(X) = $150^2 $, we solve for $ \alpha $ and $ \lambda $:
$500\lambda = \alpha$
Therefore, the variance will be:
$ \alpha/\lambda^2= 150^2$ $ 500\lambda/\lambda^2 = 22500$
$\lambda= 500/22500 = 0.0222$
$\alpha = 11.111 $
b) This part I am completely lost at.
Do we have to use the double expectation theorem here? This would give an expected mean for $S$ of $ 500*14 = 700$.
Still no clue how to calculate the variance though.