Two Sigma·Software Engineer·Technical Phone Screen·Intermediate

Intermediate

Apr 2026

Summary

Two Sigma quant engineer interview with a probability/statistics theory question that had a real proof component to it. Not the usual plug-and-chug stuff, they actually wanted rigorous justification.

Questions Asked (1)

Let X and Y be random variables and f(X, Y) a measurable function. First, is it always true that the minimum over y of E_X[f(X, y)] is less than or equal to E_{X,Y}[f(X, Y)]? Prove it or give a counterexample. Second, does this inequality hold with equality conditions when X and Y are independent? Justify rigorously.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

The first part clicked pretty fast for me.

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AI HintsAI Generated

Suggested Approach

First, interpret the inequality as a minimax-type bound: for any fixed y, E_X[f(X,y)] ≤ E_{X,Y}[f(X,Y)] is not generally true without conditions. Instead, note that min_y E_X[f(X,y)] ≤ E_X[f(X, Y)] for any random Y, and then take expectation over Y to get min_y E_X[f(X,y)] ≤ E_{X,Y}[f(X,Y)]. For the second part, analyze when equality holds, especially under independence, and discuss conditions such as f being separable or achieving the minimum almost surely.

Pro tip: Emphasize the role of measurability and integrability to avoid technical pitfalls, and connect the inequality to the probabilistic method and minimax theorems, which are common in quantitative finance and algorithmic trading interviews.

1. Clarify assumptions and notation

State that X and Y are random variables on a common probability space, f is measurable, and expectations are well-defined (e.g., f is bounded or integrable).

2. Prove the inequality

For any fixed y, E_X[f(X,y)] ≤ E_X[f(X,Y)]? Not necessarily. Instead, use the fact that for any random variable Y, min_y E_X[f(X,y)] ≤ E_X[f(X,Y)] pointwise, then take expectation over Y to obtain the result.

3. Address equality conditions

Equality holds if and only if there exists a minimizer y* such that f(X, y*) = f(X, Y) almost surely, or more generally if the minimum is achieved almost surely by Y. Under independence, equality may hold if f is separable or if the minimum is constant.

4. Provide counterexamples or edge cases

Give a counterexample where the inequality is strict, e.g., f(x,y) = (x-y)^2 with X,Y independent standard normal, then min_y E[(X-y)^2] = Var(X) = 1, while E[(X-Y)^2] = 2.

5. Discuss independence and equality

Under independence, E_{X,Y}[f(X,Y)] = E_Y[ E_X[f(X,Y)] ]. Equality in the inequality requires that the minimum is achieved almost surely by Y, which is not generally true even under independence.

Key Points to Mention

Measurability and integrability conditions
Minimax inequality and its proof via pointwise bound
Equality conditions: existence of a minimizer almost surely
Independence does not guarantee equality
Counterexample with quadratic loss
Connection to Jensen's inequality and conditional expectations

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.