Suppose the binomial pdf described in Example 5.8.2 refers to the number of votes a candidate might receive in a poll conducted before the general election. Moreover, suppose a beta prior distribution has been assigned to θ, and every indicator suggests the election will be close. The pollster, then, has good reason for concentrating the bulk of the prior distribution around the value θ = 1/2. Setting the two beta parameters r and s both equal to 135 will accomplish that objective (in the event r = s = 135, the probability of θ being between 0.45 and 0.55 is approximately 0.90).

(a) Find the corresponding posterior distribution.

(b) Find the squared-error loss Bayes estimate for θ and express it as a weighted average of the maximum likelihood estimate for θ and the mean of the prior pdf.