J We often use the bdistribution to model continuous random variables where the distribution is approximately symmetric but with a higher possibilit}F of outliers compared to a normal distribution. Consider the following course grade data for a class of size n=15 students, T8, 50, 1'2, 72, 75, 72, 58, 94, , 92, 65, 91], 54, 7'1, 45. With lower and upper outliers in the observed data, a t-distribution mall,F be more appropriate to model the distribution of course grades. Suppose y1,...,y,, are a sample from a t distribution with location parameter ,1, scale parameter or, and known degrees of freedom 1!. Assmning conditional independence given . -|.' +1112 the parameters, the likelihood mction is given by p(y1 , ..., yum, J, y) = H\" l (1 + QLJQEE) v . i=1 3 Assuming a non-informative prior 53(1': or] or 01 the posterior density;r is given by TI. 1 1 ,_ ,2 P(p,cr|y1,...,y,,,u} [x _H_ (1+ (y I) or _ or :72 1:1 ){u+ l}f2 To obtain posterior draws of a and (I, we can actually implement a Gibbs sampler. First, we need to introduce a new scale parameter A and each observation 3;, is represented as a scale mixture of normals. The new representation of the model is slams ~ Normanmh/i) lily ~ Gommo(yf2, Hf2) p{,u,.::r} ac 1,110. where H is assumed xed and known- (a) [3 marks] Reparametise the model in terms of 0'2 instead of or, and write out the joint posterior densityr of all parameters (a, :12, 31,) (i = 1, ..., a) [b] [2 marks] Derive the full conditional distribution of A, [i = 1, ...,n] given a and :72 and u. (c) [2 marks] Derive the full conditional distribution of ,1 given A, {i = 1, ...,n) and :72 and u. [d] [2 marks] Derive the full conditional distribution of or? given a and A; {i = 1, ...,a) and u