Unbiased from five days

What this unbiased estimation question tests

This is a medium-difficulty statistics problem that combines Poisson inference with the theory of unbiased estimators of minimum variance (UMVUE). It appears in quantitative interviews where candidates must move beyond plug-in estimators and reason about the properties an estimator must satisfy.

The problem gives you independent daily order counts, each Poisson distributed with an unknown rate parameter. You observe their sum and must construct an unbiased estimator for a non-linear function of the parameter—in this case, the probability of zero orders on any given day. The solution requires understanding sufficient statistics, the Rao–Blackwell theorem, and how to leverage the structure of the exponential family to derive the minimum-variance unbiased estimator.

• Sufficient statistics for the Poisson distribution
• Unbiasedness and minimum variance
• Rao–Blackwell theorem and UMVUE construction
• Conditional expectation given a sufficient statistic