Suppose n 3 and observations are 2 5 and 7 using

Question

Question 1 - Suppose we have n independent observations from

P(y|θ) = θ/y², y > θ, and θ > 0

(a) Find a conjugate prior distribution for θ.

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

(c) Suppose n = 3 and observations are 2; 5 and 7. Using simulation, find the posterior mean and variance for θ and compare your results with (b).

(d) Using simulation and the observations in (c), find a 95% credible set for θ.

Question 2 - Suppose that your prior for θ is a ¼  to ¾ for a mixture of N(0, 1) and N(1, 1) and that a single observation y∼N(θ, 1) turns out to equal 3. What is your posterior probability θ > 2?