2.4 Bickel and Mallows (1988) further investigate the relationship between unbiasedness and Bayes, specifying conditions under which these properties cannot hold simultaneously. In addition, they show that if a prior distribution is improper, then a posterior mean can be unbiased. Let X ∼ 1

θ f (x/θ), x > 0, where ∞

0 tf (t)dt = 1, and let π(θ) = 1

θ2 dθ,

θ > 0.

(a) Show that E(X|θ) = θ, so X is unbiased.

(b) Show that π(θ|x) = x2

θ3 f (x/θ) is a proper density.

(c) Show that E(θ|x) = x, and hence the posterior mean, is unbiased.