In exercises 2b and 2c and 3d there are some misprints.
In the last sentence of 2b, you should derive the posterior distribution for sigma.
In 2c you should derive the marginal distribution if theta is fixed, not sigma.
In 3d, the phi in the conditional distribution should be psi.
For model 2 in ex 3 the Poisson model should also have exp(theta_t)
In the table before 3e, the estimates for beta_0, beta_1, beta_2 should be
model1: ...