Compound risk poisson models

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.

EDIT

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.

Yes, if I understand you correctly. The law of total expectation, or law of iterated expectation, says in this case that $$\mathbb{E}[S] = \mathbb{E}[\mathbb{E}( S \mid N) ].$$ To find the variance, there's an analagous formula called the law of total variance, which says $$\mathbb{V}[S] = \mathbb{E}[\mathbb{V}( S \mid N) ] + \mathbb{V}[\mathbb{E}( S \mid N) ].$$ Let me know if that helps.