Please solve this question precisely

Suppose that Y is a random variable belonging to a special subset of the exponential family where the density function of Y' has the form f(y.0,4) =exp -23+ c(y.) P For some constants 0 and @ and functions b and c. () Show that the moment generating function of Y is given by My (t) - exp [b(0+to)-b(0) [3] Hint: Note that the function f(y, 0 - pr, ) is the density of another random variable of the same family and hence ( f(y.0+qq)dy =1. (ii) Show that E(Y) = b'(0) and Var(1) = qb'(0) using the result in (i). [4] (iii) Verify that the result in (i) holds if Y has a Poisson distribution. [4] [Total 11] Claims on a portfolio of insurance policies arise as a Poisson process with parameter 2. Individual claim amounts are taken from a distribution X and we define m; = E(X) for i - 1, 2, .... The insurance company calculates premiums using a premium loading of 0. (i) Define the adjustment coefficient R. [1] ) () Show that R can be approximated by 20171 by truncating the series expansion of My() (b) Show that there is another approximation to R which is a solution of the equation maps + 3may - 60m, = 0. [6] Now suppose that X has an exponential distribution with mean 10 and that 0 = 0.3. (iii) Calculate the approximations to & in (ii) and (iii) and compare them to the true value of R. [6] [Total 13]