Assume that a policyholder is five times more likely to file exactly four claims as to file exactly two claims. Assume also that the number $X$ of claims of this policyholder is Poisson. Determine the expectation $E((X+1)^2)$.
I solve for Poisson and got $\lambda=\sqrt{60}$ but don't really understand where to go from there.
Just use the linearity of the expectation, plus the facts that $Var(X)=E[X^2]-E[X]^2$ and, for the Poisson distribution, $Var(X)=E[X]=\lambda$:
$$ E[(X+1)^2]=E[X^2]+2E[X]+1=Var(X)+E[X]^2+2E[X]+1=\lambda^2+3\lambda+1 $$