Suppose that X is Bernoulli random variable with P(X=1)= and P(X=0)=1 , and it is desired to guess the value of on the basis of X under the squared error loss function L(,d)=(d). Assume that the domain of is =[91,98].

(a) Assume that has a prior distribution on two endpoints of with probability mass function a(=91)=21 and a(=98)=21 . For the corresponding Bayes procedure, denoted by a, show that its Bayes risk ra(a)<supRa(), and thus this direction does not work.

(b) Consider another prior distribution b(=422)=21 and b(=42+2)=21, which is a well-defined prior over =[91,98], since 91<422<42+2 <98 . Show that the corresponding Bayes procedure, denoted by b, satisfes rb(b)=supRb(), and thus we can conclude that b is minimax on =[91,98].