Suppose the true parameter value is θ*. What is the expected value of the Bayes estimate, in terms of θ*, if we obtain a random sample of size N?
  • E(θbayes) = E((n+1) / (n+m+2)) by part (c)
    = E((n+1) / (N+2)) since n+m = N is constant
    = (E(n) + 1) / (N+2) by linearity of expectation
    = (Nθ* + 1) / (N+2) since n ~ Binomial(N, θ*)
    = θ* + (1-2θ*) / (N+2)

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