15. Suppose that Y | G(1, ) (i.e., the exponential distribution with mean ), and that...

15. Suppose that Y |θ ∼ G(1, θ) (i.e., the exponential distribution with mean

θ), and that θ ∼ IG(α, β).

(a) Find the posterior distribution of θ.

(b) Find the posterior mean and variance of θ.

(c) Find the posterior mode of θ.

(d) Write down two integral equations that could be solved to find the 95% equal-tail credible interval for θ.

16. Suppose X1,...,Xn iid

∼ NegBin(r, θ), r known, and that θ is assigned a Beta(α, β) prior distribution.

(a) Find the posterior distribution of θ.