← Squarepoint Interview Insights
My first instinct was to think about a uniform distribution, which felt wrong almost immediately.
Start by clarifying that the interval is fixed and the random variable must take values within it. Then use the variance formula and properties of expectation to show that the maximum variance is achieved by a two-point distribution at the endpoints, with equal probabilities. Finally, discuss the intuition and any assumptions.
Pro tip: Mention that this is a classic result in probability: for a bounded random variable, the variance is maximized by pushing all mass to the extremes. This demonstrates deep understanding and connects to real-world risk modeling.
Confirm that the random variable must take values in [a, b] and that we want to maximize its variance. Ask if there are any other constraints.
Use the formula Var(X) = E[X^2] - (E[X])^2. Note that for fixed mean, variance is maximized by spreading mass as far from the mean as possible.
Show that the maximum variance is (b-a)^2/4, achieved by a Bernoulli-like distribution: P(X=a)=P(X=b)=1/2. Prove via Jensen's inequality or by considering the convexity of the variance as a function of the distribution.
Explain that this is the most 'spread out' distribution possible on the interval. Mention that any other distribution will have smaller variance because it places mass inside the interval.
If time permits, mention that if the mean is fixed, the maximum variance might be different, or if the distribution must be continuous, the supremum is still the same but not attained.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.