X₁,X₂ are iid uniform(0,θ),θ>0, with maximum T=X_(2). Let ψ(θ)=P_θ(X₁+X₂<1). A continuous δ:(0,∞)→ℝ makes δ(T) an unbiased estimator of ψ(θ) for every θ. Find 18δ(3/4) (integer).
Rao–Blackwellize the indicator of X₁+X₂<1 given T=t. The other observation is uniform on (0,t), so δ(t)=1 for t≤1/2, (1−t)/t for 1/2<t<1, and 0 for t≥1. The maximum is complete: if ∫₀^θg(t)2t/θ²dt=0 for every θ, differentiation forces g=0 almost everywhere; continuity fixes point values. Thus the given continuous unbiased estimator must be this δ. At t=3/4, δ=1/3, giving **6**.
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