The Borel Paradox (Miscellanea 4.9.3) can also arise in inference problems. Suppose that X1 and X2 are iid exponential(θ) random variables.
a. If we observe only X2, show that the MLE of  is θ = X2.
b. Suppose that we instead observe only Z = (X2 - 1)/X1. Find the joint distribution of (X1 ,Z), and integrate out X1 to get the likelihood function.
c. Suppose that X2 = 1. Compare the MLEs for 9 from parts (a) and (b).
d. Bayesian analysis is not immune to the Borel Paradox. If π(6) is a prior density for θ, show that the posterior distributions, at X2 = 1, are different in parts (a) and (b).