Collective Risk Model

Question

So I am asked a specific question about the collective risk model S=X1+...+XN where N is Bin(3,q=2/3) and the claim size is uniformly distributed on (1,2,3).

I am asked to calculate E(S), Var(S) and fs(r).

Now the problem I am having is that I have literally no real examples to go on. I am having difficulties understanding firstly the E(S). I know that E(S) = E(N)E(X)and E(N) = nq but what is n here ? Is it 3 or is it 2 from the uniform ? Really unsure. I know how to calculate the binomial probability but again just unsure on how to calculate

For Var(S) again I know the formula is E(n)Var(X) + Var(N)E^2(X) but again I have no examples to go on as to how to calculate these things.

Then for the Fs(r) I have absolutely no clue how to get the alpha and beta values.

If anyone could either do this out for me or link me to a page with a few examples to practice I'd appreciate it !

Thanks

Answer 1

For the expectancy of the random variable $S$, we can use the partition theorem. We know that $\mathbb{E}(S)=\sum_{i=0}^{3} \mathbb{E}(X_1+\dots+X_n \mid N=i)\mathbb P(N=i)$. Now, it holds that (as $X_1,\dots,X_n$ are independent)$ \mathbb{E}(X_1+\dots+X_n=\mathbb{E}(X_1)+\dots+\mathbb{E}(X_n)=N\cdot \mathbb{E}(X_1)=N\cdot \left(\frac{1+2+3}{3}\right)=2N$ .

So, using this we have that: $\mathbb E(S)=\sum_{i=0}^{3} \left(2i\cdot \binom{3}{i}\left(\frac{2}{3}\right)^i\left(\frac{1}{3}\right)^{3-i}\right)$.

Now, can you finish it?