Suppose X1,,Xn are the indicators of n Bernoulli trials with success probability . And suppose l(,a) is the quadratic loss (a)2 and the prior () is the beta, (r,s), density.

• Find the Bayes estimate ^B of and write it as a weighted average w0+(1w)X of the mean 0 of the prior and the sample mean X=S/n.

• Show that ^B=(S+1)/(n+2) for the uniform prior.