Intermediate Statistical Theory: Bayesian estimation

Problem 3 Let Xq,...,X,, be an i.i.d. sample from the normal distribution with mean zero and variance o2 > 0. (a) Write the pdf of the distribution in terms of the parameter 7 = 1/02. [1] (b) Find the posterior distribution of the parameter 7, using a gamma(c, 3) prior distribution. Here, a > 0 is the shape, and 8 > 0 is the scale parameter. [4] (c) Is the gamma distribution a conjugate prior in this example? Justify your answer. [1] (d) Find the Bayes estimate of 7. You can use the result that, if Z ~ gamma(a, ), then E(Z) = af (you don't need to prove this). [2] (e) How does the Bayes estimate simplify (approximately) when n is large? Describe the connec- tion to the maximum likelihood estimate of o2. [1]