An unbiased Poisson probability
A sample of n=4 Poisson(\theta) observations sums to T=5. Using the conditional binomial distribution of one observation given the sum, compute the UMVU estimate of \mathbb{P}(X_1=1) and the maximum likelihood estimate of the same probability.
Worked answer
Given T=5, one observation is \operatorname{Bin}(5,1/4). The UMVU estimate of \mathbb{P}(X_1=1) is
\binom{5}{1}\Bigl(\frac14\Bigr)^1\Bigl(\frac34\Bigr)^4=\frac{5\cdot 81}{1024}=\frac{405}{1024}\approx 0.3955.
The maximum likelihood estimate uses \widehat\theta=5/4 in e^{-\theta}\theta:
e^{-5/4}\cdot\frac54\approx 0.3581.
The two functions of the same sufficient statistic are not equal. Only the binomial probability is unbiased for e^{-\theta}\theta.