Largest Variance on [-1, 1]

This classic quant prep question is about how much uncertainty a bounded random variable can have. You work with a variable that lives in a fixed finite range and ask how "spread out" it can be in terms of its variance. It lives at the intersection of probability theory, optimization over distributions, and basic measure-theoretic intuition, which are central themes in quantitative finance interviews.

It trains your understanding of variance as a functional of a distribution, your intuition for extremal distributions under simple constraints, and your ability to reason with expectations without writing down a density. It also sharpens comfort with symmetry, support constraints, and how distributional shape affects dispersion.

This matters for quant interviews because similar ideas appear in risk bounds, worst-case scenarios, and model calibration. Interviewers use it to see if you can reason rigorously about distributions, not just compute formulas.

What it tests

Variance measures the average squared deviation from the mean, so to maximize variance for a random variable restricted to an interval, you must place as much probability mass as possible at the points furthest from the mean. This is because the squared distance grows rapidly as you move away from the mean, so concentrating probability at the endpoints of the allowed range maximizes the sum of squared deviations. The maximum is achieved when the distribution is 'bimodal' at the extremes, and the mean is centered between them. This principle holds regardless of the interval: for any bounded variable, the variance is maximized by a two-point distribution at the endpoints, each with probability $1/2$. This is a consequence of convexity: the function $x^2$ is convex, so its expectation is maximized by extremal values.