please help me solve this clearly.

(i) Explain what a conjugate prior distribution is. (ii) The random variables X1: X2, ..., X, are independent and have density function fri = lexx (r>0) Show that the conjugate prior distribution for > is a Gamma distribution. [3] (iii) (a) The density function of 2 is AN) = T(a) Show that E(1/)) = s/(a - 1). (b) Hence if X], X2, ..., X,, is an independent random sample from an exponential distribution with parameter ), show that the posterior mean of 1/2. can be expressed as a weighted average of the prior mean of 1/2. and the sample average. 151 (iv) An insurer is considering introducing a new policy to provide insurance against the failure of toasters within the first five years of purchase. Alan and Reatrice are underwriters working for the insurer. Based on his experience of similar products, Alan believes that toasters last three years un average. Beatrice believes that six years is the average lifetime. Both are adamant and are prepared to express their uncertainties about the average lifetime in terms of standard deviations of six months and one year respectively. They decide to resolve their differences by testing a sample of toasters large enough to ensure the difference in their posterior expectations for the average lifetime will be less than one year. Calculate how many toasters they should test. assuming the exponential distribution is a good model for toaster lifetimes. You may use the fact that if A ~ [(u, s) then Van(1/) = [E(1/)] x [8] U-2 [Total 18]