suppose that we want to estimate the parameters sada theta of the geometric dist

Question

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.

Explanation / Answer

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

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