suppose that we want to estimate the parameters sada theta o

suppose that we want to estimate the parameters sada theta of the geometric distribution on the basis of a single observation. If the loss function is given by

L(d(x), theta) =c{d(x)- theta}²

and Capital theta is looked upon as a random variable having the uniform density h(theta)=1 for 0<theta<1 and h(theta)= 0 elsewhere, show

A) the conditional density of capital theta given X=x is

f(theta l x) = x(x+1)theta(1-theta)^x-1 for 0<theta<1 and 0 elsewhere

B) the Bayes risk is minimized by the decision function

D(x)= 2/(x+2)

Hint: make use of the fact that the integral of any beta density is equal to one.

Yeah...I have no idea.

Solution

A) the conditional density of capital theta given X=x is