Suppose that X is a exponential random variable with parameter 0;

that is, its probability density function is given by

and let n(0) be Gamma(a,P).

(a) Find the Bayes estimator of 0 with the squared-error loss.

(b) Let X u...,Xn be independent, identically distributed exponential random variables with parameter 0.

Show that the posterior density of 0 given X x — x u ...,Xn = xn is Gam ma(a + n, (3 + 2?=1 x t).

fx(x|0) Be 10. x>0 otherwise