Unbiased from one day

What this unbiased estimation problem tests

This is a hard statistics problem that combines theoretical understanding of unbiased estimators with practical problem-solving. It appears frequently in quant interviews because it requires you to think deeply about how to construct estimators that satisfy an exact mathematical constraint, rather than just applying a standard formula.

The core challenge is finding a function of your data that has the correct expected value for all possible values of the unknown parameter. This tests whether you can work backwards from the target (the probability of zero arrivals over two days) to build an estimator from scratch, and then evaluate it at a specific observed value. The problem rewards careful reasoning about moment-generating functions, indicator variables, and the algebraic structure of exponential families.

• Properties of the Poisson distribution and its moments
• Construction of unbiased estimators via moment equations
• Uniqueness of unbiased estimators and their connection to the data
• Numerical evaluation under constraints