Suppose that 100 scores are independent to each other and they follow Normal distri- bution with mean = 75 and unknown precision 7. Specifically, let 21, . .., Tion be the final scores, and ri ~N(75, -) for i = 1, ..., 100. Suppose that the precision + has a Gamma(2, 1) prior distribution, where Gamma(o, B) has the pdf f ( I) = T( 0 ) "-1 exp (-Bx). (a) (5 marks) Derive the posterior distribution of the precision 7 conditioned on the final exam scores of 100 students, p(7 1, . . .; $100)- (b) (6 marks) Derive the posterior predictive distribution for a new score i, p(i|71, . . ., $100). Hint for (b)] A three-parameter version of a t distribution (Jackman, S. (2009)), denoted by t(v, a, b), has the pdf p(x|v, a, b) = 1 + 1(x - a)2\\ TC)VTub V