I found the questions a bit tricky. help.

Explain why the link function g(#) =log / is appropriate for the Poisson distribution by considering the range of values that it results in /, taking. 2 Explain the difference between the two types of covariate: a variable and a factor. 3 (i) A random variable Y has density of exponential family form: fly)= exp ye-b(@) + c(y.$) a($) State the mean and variance of Y in terms of b(0) and its derivatives and a() . [1] (ii) (a) Show that an exponentially distributed random variable with mean / has a density that can be written in the above form. (b) Determine the natural parameter and the variance function. [3] [Total 4] An insurer wishes to use a generalised linear model to analyse the claim numbers on its motor portfolio. It has collected the following data on claim numbers y;, i=1, 2, .., 35 from three different classes of policy: Class I 1 2 0 2 1 0 0 2 2 1 Class II 1 0 Class III 0 0 0 1 0 1 0 0 1 0 H 0 0 0 0 0 0 0 For these data values: Evi =11 Ey1 =3 35 Ey =4 i-1 i-11 i-16 The company 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]