Compute the value of the marginal density

Question

Solve the following problem:

Starting from the prior distribution

β|σ²,X∼N_k+1(β¯, nσ²(X^TX)^-1):

a. Show that

Xβ|σ²,X∼N_n(Xβ¯, nσ²(X^TX)^-1X^T):

and that

y|σ²,X∼N_n(Xβ¯, nσ²(I_n+nX(X^TX)^-1X^T).

b. Show that integrating in σ² with π(σ²)=1/σ² yields the marginal distribution of y above.

c. Compute the value of the marginal density of y for the swiss dataset.