Solution
Rao-Blackwell for p Squared
Show the problem again
Four independent Bernoulli trials with success probability p = 1/2 are observed, X₁, ..., X₄. The crude estimator X₁X₂ is unbiased for p². Let T = X₁ + X₂ + X₃ + X₄ be the total number of successes, which is sufficient for p, and let W = E[X₁X₂ | T] be the Rao-Blackwellised estimator. Find Var(W) when p = 1/2. Give a reduced fraction.
Worked solution
Given T = t, the successes are placed uniformly among the four positions, so P(X₁ = X₂ = 1 | T = t) = t(t − 1)/(4·3). Thus W = T(T − 1)/12, which is unbiased for p² (E[T(T − 1)] = n(n − 1)p²). With T ~ Binomial(4, 1/2), the probabilities of T = 0..4 are 1,4,6,4,1 over 16 and W takes the values 0, 0, 1/6, 1/2, 1. Then E[W] = (6/6 + 4/2 + 1)/16 = 1/4, and E[W²] = (6/36 + 4/4 + 1)/16 = (13/6)/16 = 13/96. Hence Var(W) = 13/96 − 1/16 = 7/96. The crude estimator X₁X₂ has variance 1/4 · 3/4 = 3/16 = 18/96, so conditioning on the sufficient statistic cut the variance by more than a factor of 2.5. The answer is 7/96.
Source: Original problem written for AxiomIQ, applying the Rao-Blackwell theorem.