Question

Let X₁, ..., X_n be a random sample of an Exponential population with parameter p. That is,

f(x|p) = pe^-px , x > 0

Suppose we put a Gamma (c, d) prior on p.

Find the Bayes estimator of p if we use the loss function L(p, a) = (p - a)².

Answer 1

Let X₁, ..., X_n be a random sample of an Exponential population with parameter p, i.e.,

The likelihood function is

The prior distribution of p follows Gamma (c,d), i.e.,

Then, the posterior distribution is the product of likelihood and prior distribution

Under the squared loss function L(p, a) = (p - a)² , Bayes estimator is

The integral is a gamma function and its density is equal to one when we multiple by some constant, consider

a = n+c+1

Then,