Unbiased Estimation of a Zero Probability

Four independent observations X₁, ..., X₄ come from a Poisson distribution with unknown mean λ. The quantity of interest is e^(−λ) = P(X₁ = 0). The crude estimator 1{X₁ = 0} is unbiased. Rao-Blackwellise it by conditioning on the sufficient statistic T = X₁ + ... + X₄. If the observed total is T = 5, what is the value of the resulting estimator? Give a reduced fraction.

• bias
• combinatorial games
• distributions
• estimation
• variance

Source: Standard Rao-Blackwell / UMVUE example for the Poisson family; statement written for AxiomIQ.