Please help me tackle the following question. Thanks

The random variable S represents the annual aggregate claims for an insurer from policies covering damage due to windstorms. S is modelled as follows: where: M denotes the number of windstorms each year and has a Poisson distribution with mean k denotes the aggregate claims from the ith windstorm and is modelled as Y= Xi where: N; denotes the number of claims from the ith windstorm. N, N2.... , NM are independent and identically distributed random variables, each with a Poisson distribution with rate 2. Xji denotes the amount of the /th claim from the ith windstorm. X;;, i= 1, ..., M,j= 1, ..., N; is a sequence of independent and identically distributed random variables, each with mean H and variance o'. It is assumed that the random variables M, N; and X,; are independent of each other. (i) Derive expressions for the mean and the variance of Y, in terms of 2, p and s. [2] (Li) Derive expressions for the mean and the variance of $ in terms of k, ), u and o. (3] Now suppose that X, has an exponential distribution with mean 1. (iii) Show that for any positive numbers r and C P(X; Sx+ CIX; > C)= P(X, Sx). (3]Consider the new random variable S, given as: M N 1=1 / = 1 where: Xy = |Xj-2 if X;; 22 otherwise Let W, be the number of non-zero X,, amounts, i.e. the number of claim amounts from the ith windstorm that arc greater than 2. Also assume that M N2.....Nyr are independent and identically distributed Poisson random variables, with parameter A . Let k = 4. ) = 1,000. (iv) (a) Show that A" = 135.3. (b) Explain why the distribution of X;; is exponential with mean 1. (c) Calculate the mean and variance of SR- [7] [Total 15]