Consider the following Bayesian model: f(, 0) X,...,n | ,0 iid N(, 0). 1/0, R, o > 0, (i) Derive an expression for the joint posterior pdf of (, o) given x = (x,...,xn). (ii) Show that n i=1 n (x; ) = (x; x) + n( ) . - i=1 [2 marks] [2 marks] (iii) Using part (ii), or otherwise, show that the conditional posterior f(ulo, x) is normal. Find the mean and variance of this normal distribution. [2 marks] (iv) Show that the conditional posterior f(o|u, x) is Inverse-Gamma. Find the shape and scale of this Inverse-Gamma distribution. [2 marks] (v) Describe how you would sample from the joint posterior f(u, o|x). [4 marks] (vi) Describe how you construct posterior 90% credible intervals for the mean parameter and the variance parameter o. [2 marks]