INFERENCE STATISTICS

cuesnou a [32] Suppose that n individuals are chosen from a large population with an unknown proportion 11 of H I V\" positive individuals. Furthermore, suppose that the prior distribution of or. has prior probability densityr function 1 if [i g a: 5 1, Elf-9:) = E] otherwise. {a} Prove or disprove that the posterior distribution of a: is a home: + 1, n x -l- 1) distribution with probabilityr density function F\"! +2} or ftlx} T(x+]}f[n x+]) "(1 of\" ifO g a 1:]. otherwise. {Ti Facts which you may use: A random random variable Y has a beraia, 1:) distribution if it has probability density' function: Ffa +b} l :51 . 1:] otherwise. 1 H" + 3'} o'l hl {1 d : 1- A 1"(a)1"(b}y 3") y {b} Find the mean (E). the mode (a), and the variance of the posterior distribution in part la}- [3+4+4} (c) Compare the risk functions of the mean {a} and the mode {6:} for the case when n = 2. Which is a better point estimator of a? {14}