Answer all the questions

1 Let 0 -binomial(N, q), where q is unknown and A is large. Using the fact that the posterior distribution of q is approximately N(4, V), where -1 V = doz (log D)(4) find a formula for an approximate 95% Bayesian confidence interval for q. 2 The following table gives the exposed to risk, Exo, and deaths 640, for certain male and female tuberculosis patients in Sweden: exposed to risk deaths males 221 68 females 194 83 Test, at the 1% significance level, the hypothesis that the true rate of mortality at age 40 is the same for both sexes. 3 The following data refer to surrenders among the holders of three types of life assurance contract (mortality being ignored.) No. of policies in force No. of surrenders Policy type on 1 January 2000 in 2000 A 2,489 167 B 1,524 93 C 3,002 173 Test the hypothesis that the true surrender rates are the same for all 3 types of contract.The random variable Y is defined by: N Y =>X; i=1 where the X,'s are independent Bin(n, p) random variables and N is a Poisson random variable, independent of the X, 's, with mean 1. (i) Prove from first principles that the moment generating function of Y is given by: My(t) =exp a[(q + pe' )" -1]} [4] (ii) Hence derive expressions for the mean and variance of Y . [4] [Total 8]