Maximal maximum

What this hard probability-bounds interview question tests

This is a hard question in statistics and data analysis that combines moment constraints with optimization. It asks you to find the tightest upper bound on a point probability given only the mean, variance, and support of a distribution—a skill quant researchers and traders use when reasoning about tail risk and extreme outcomes under partial information.

The challenge lies in recognizing that you must construct the distribution that maximizes the probability at a specific point while respecting the given constraints. This requires setting up the optimization problem correctly, identifying which moment constraints are binding, and solving for the probability mass allocation. Strong answers explicitly state the distribution that achieves the maximum and verify that it satisfies all given conditions.

• Moment constraints and their role in bounding distributions
• Optimization over the space of probability measures
• Support bounds and their interaction with variance
• Chebyshev and moment-based inequalities