Hi please help me with the following questions Thanks Let X1

Hi, please help me with the following questions. Thanks!

Let X1,...., Xn be iid observations from a beta(theta, 1) pdf and assume that theta has a gamma(r, lambda) prior pdf. Find a 1-a Bayes credible set for theta.

Solution

X~(theta,1), then f(x)=(1/(theta,1))*x^(theta-1), theta~gamma(r,lambda) then f(theta)=[(e^(-rtheta))(theta^(lambda-1)(r^(lambda)))/gamma(lambda),then the bayes theorem is f(theta/x)=(f(x/theta)f(theta))/f(x) proportional to (1/(theta,1))*x^(theta-1)*[(e^(-rtheta))(theta^(lambda-1)(r^(lambda)))/gamma(lambda). A 100(1-theta)% confidence interval is an actual interval that contains the parameter theta with probability (1-theta), then P(theta_l<theta<theta_u/x)=(1-theta) and P(theta<theta_l/x)=theta/2= P(theta>theta_u/x).