Questions about posterior linear regression

Just 2 simple questions I´m struggling with. Hope you can help.

Suppose that the model $y=\beta X +\epsilon$ with $\epsilon \sim \text{ Normal}(o,\sigma^2I_n)$ has a prior $\beta \sim \text{ Normal}( \beta_0, k(X^TX)^{-1})$

I want two things:

1.) I want to show that for the density of $\beta$ that

$$p(\beta) \propto \exp \{-\frac{1}{2} k^{-1} (\beta^TX^TX\beta-2\beta^TX^TX\beta_0) \}$$

2.) I want to show that $$p(\beta | y,X,\sigma^2) \propto p(y|X,\beta, \sigma^2)\cdotp(\beta)$$

It would be very nice if somebody can say anything about this. I would also know why $$p(\beta|y,X,\sigma^2) \propto \exp\{-\frac{1}{2}(k^{-1}+\sigma^{-2})\beta^TX^TX\beta -2\beta^TX^TX\beta_0 \}$$

• If that is a homework, then add a "self-study" tag. – Stat Oct 22 '13 at 11:30
• Is $\sigma^2$ assumed to be known? – Stijn Oct 22 '13 at 12:44
• @Stijn yes .... – Applied mathematician Oct 22 '13 at 14:03