Oct 08, 2026 by Offer

Citadel Normal Variables Question 2027: Expected Maximum

The Citadel expected value normal variables question — commonly reported by candidates as "What is the expected value of the maximum of two standard normal variables?" — has the answer 1/√π ≈ 0.564. The derivation uses one identity: max(X, Y) = (X + Y + |X − Y|) / 2.

What This Citadel Expected Value Normal Variables Question Assesses

This tests whether you can decompose an unfamiliar expectation into pieces you know. Citadel isn't checking if you've memorized the answer — it's checking whether you reach for the max-identity, recognize the distribution of the difference, and know the absolute moment of a normal. Each step is standard; chaining them under pressure is the skill.

How to Answer This Citadel Expected Value Normal Variables Question

Derive it live in four steps:

Step 1 — Apply the identity. max(X, Y) = (X + Y + |X − Y|)/2. Take expectations: E[max] = (E[X] + E[Y] + E|X − Y|)/2 = E|X − Y|/2, since both means are zero.
Step 2 — Find the distribution of the difference. X − Y is normal with mean 0 and variance 1 + 1 = 2, i.e., X − Y ~ N(0, 2).
Step 3 — Absolute moment of a normal. For Z ~ N(0, σ²), E|Z| = σ√(2/π). With σ = √2: E|X − Y| = √2 · √(2/π) = 2/√π.
Step 4 — Halve it. E[max(X, Y)] = (2/√π)/2 = 1/√π ≈ 0.564.

Example line: "Using max equals the average plus half the absolute difference, and the difference is N(0,2), I get 1 over root pi — about 0.56."

Common Mistakes

Guessing zero by symmetry. Symmetry gives E[max] = −E[min], not zero — the maximum of two draws skews positive, which is exactly the point.
Forgetting the variance adds. X − Y has variance 2, not 1; using σ = 1 gives 1/√(2π) ≈ 0.399, the most common wrong answer.
Trying to integrate the joint density directly. It works but wastes minutes — the identity is the intended shortcut, and interviewers watch for it.

Candidates commonly report this as a Citadel question where the interviewer cares more about the path than the number. If you blank on E|Z|, say so and derive it or bound it — showing structured recovery beats silent stalling.

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FAQ

What's the expected minimum? By symmetry, −1/√π ≈ −0.564. The pair (max, min) is symmetric around zero.

Does this generalize to n variables? Yes, but there's no closed form — E[max] of n standard normals grows like √(2 log n). The n = 2 case is special in having the clean 1/√π answer.

What if the variables are correlated? The identity still holds, but X − Y then has variance 2(1 − ρ), so the answer becomes √(1−ρ)/√π. Positive correlation shrinks the expected maximum.

Why does the identity max = (x+y+|x−y|)/2 hold? If x ≥ y, the right side is (x + y + x − y)/2 = x. If y > x, it's (x + y + y − x)/2 = y. It just picks the larger one.

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