Please help solve the following microeconomic question. Thank you

(i) A random variable Y has density of exponential family form: vo-b(0 ) f ()) = exp + c(),0) a(g) State the mean and variance of Y in terms of b(0) and its derivatives and a(6). (ii)(a) Show that an exponentially distributed random variable with mean / has a density that can be written in the above form. (ii)(b) Determine the natural parameter and the variance function. 131 [Total 4]An insurer wish es to use a generalised linear model to analyse the claim numbers on its motor portfolio. It has collected the following data on claim numbers yi, i =1, 2, .., 35 from three different classes of policy: Class I 2 0 2 0 0 2 2 Class II 1 0 1 - Class III 0 0You are given that: 10 15 35 Zy =4 i=l 1=16 It wishes to use a Poisson model to analyse these data. (i) Show that the Poisson distribution is a member of the exponential family of distributions. [2] (ii) The insurer decides to use a model (Model A) for which: i = 1, 2, ..., 10 log A; = B i =11, 12. .... 15 i = 16, 17, ..., 35 where /, is the mean of the relevant Poisson distribution. Derive the likelihood function for this model, and hence find the maximum likelihood estimates for a, B and 7 . (4] (iii) The insurer now analyses the simpler model log /, = 0, i =1,2, ..., 35, for all policies. Find the maximum likelihood estimate for a under this model (Model B). [2] (iv) Show that the scaled deviance for Model A is 24.93, and find the scaled deviance for Model B. [5] You can assume that f(y) = ylogy is equal to zero when y = 0. (v) Compare Model A directly with Model B, by calculating an appropriate test statistic. [2] [Total 15]