Please solve

1. van den Bergh [1985] considers the luminosity for globular clusters in various galaxies. In the paper, vdB's conclusion is that the luminosity for clusters in the Milky Way is adequately de- scribed by the N(#, o') distribution, where a (E R) represents the population average brightness and o (> 0) is the population standard deviation of brightness. Its PDF is f(x | 0) = f(z | p, o') = - 1 - exp (x - p)2 V2no 202 for r E R. Given the observed data of n globular clusters r = {$1, 12, ..., In}, we are interested in esti- mating a using a Bayesian approach when o' is unknown. As shown in the class, a frequentist way is to replace of with the sample variance $2, making the pivotal quantity X - / S/ vn follow the to-1 distribution. The resulting confidence interval based on this pivotal quantity becomes wider to account for additional uncertainty of not knowing of (i.e., of replacing o' with an estimator). On the other hand, a Bayesian way of handling an unknown parameter (that is not of interest) is to integrate it out from the joint posterior density function. For a Bayesian inference on the vdB's problem, we assume a non-informative joint prior density for / and of as follows to express our lack of knowledge about these two parameters: f(#, 0?) x -(NER, 0230); or more simply f(A, a?) x- (a) (4 points) Find the marginal posterior density function of /, i.e., Hint 1: Be careful about a constant term needed when you integrate out of from f(u, o' | x). To use the fact that an integration of any PDF is 1 (the area under any PDF is 1), an integration of the essential part of a PDF requires multiplying by some constant to make the essential part (integrand) the actual PDF; see the example of deriving the marginal posterior of o' covered in the class. The functional form of the inverse-Gamma(o, 8) PDF will be helpful. Hint 2: Do not apply the following equation before you integrate out a?. Hint 3: After integrating out o', the following equation will be helpful. C(x-) = (x -1)+nu - x) =(n -1)s' +n(u - 2)? = (n- 1)83 (1+ 1(1-1)2) 1=1 (n - 1)$2