Hi, I was wondering if you could help me with a problem of understanding a solution. In the notes of the ruin theory chapter it states that: S = X_1 + X_2 + X_3 etc and E(S) = lambda * E(X) Var(S) = lambda * E(X^2) Okay I think i understand this. But when i come to do the question in Section 2.4, page 14/15, I see that the variance calculated for S(1) is actually lambda * Var(X), why is this? Is it because of the equation on 16 of the acty tables: Var(S) = E(N)var(x) + var(N)E(X)^2 and we are assuming that the var(N) = 0? Why do we assume this? Any help would be greatly appreciated.
Hi, I think it's because the initial formulae you quote are for the special case where N is a Poisson process with parameter lambda. If N ~ Poi(lambda) then E(N)=var(N)=lambda. Plugging that into the equation from the tables gives var(S)=lambda var(X) + lambda E(X)^2 = lambda E(X^2) from before. If N were some other distribution then this wouldn't necessarily be true
