1. Let S = _ X; be the loss, where N is the frequency distribution and X is the severity distribution. The mof of S is Ms(t) = MN(In Mx (t) ) where My and My are the mgfs of N and X respectively. a. Prove, by differentiating Ms using the chain rule and known properties of mgfs, that E(S) = E(N) E(X) and var(S) = E(N)var(X) + var (N)E(X) 2. Hint: for the second derivative (fgh)'=f'gh+fg'h+fgh' b. Now let X be Exponential with mean S and let N be Poisson with rate parameter 2. Give the form of Ms as a function of t (i.e. substitute for the two mgfs), and show that Ms(t) = ABMs(t) (1 -Bt) . Hence or otherwise, derive the mean and variance of S using this mgf. c. Again let X be Exponential with mean S and let N be Poisson with rate parameter . Evaluate the mean and variance of S using the formulae given in (a), and compare to those found in (b)