Inference Statistics

Question 3 [32] Suppose that n individuals are chosen from a large population with an unknown proportion o: of HI [1+ positive individuals. Furthermore, suppose that the prior distribution ofcx. has prior probability density function 1 if C- 5 a g 1, 3(a) : 6 otherwise. (a) Prove or disprove that the posterior distribution of a: is a berafx +1, :1 x +1) distribution with probability density function rm +2} n; +1}r(n x +1)\": "(1 a?\" ift) 5 o: 5 1. fish) U otherwise. (7} Facts which you may use: A random random variable Y has a beta[o, in) distribution if it has probability density function: l(' +3?) {11 451 . 0 otherwise. 1 Na + 3'} al 211 (1 d = 1. A l'lfa)l'(b}y y) y (b) Find the mean (a). the mode (a), and the variance of the posterior distribution in part (a)- (3+4+4} {c} Compare the risk functions of the mean {E} and the mode {ii} for the case when n = 2. Which is a better point estimator of a? (14}