In a simple linear regression problem with the usual improper prior, prove that the conditional mean of 0 given 1 is Use the fact that

In a simple linear regression problem with the usual improper prior, prove that the conditional mean of β0 given β1 is Use the fact that (β0, β1) has a bivariate normal distribution as described in Theorem 11.4.1, and then use Eq. (5.10.6) to find the conditional mean.