Citadel·Software Engineer·Technical Phone Screen·Intermediate

IntermediatePrefer not to say

Apr 2026

Summary

Citadel quant engineer interview with a pretty dense probability/statistics question that required knowing some linear algebra in the background. The follow-up was sneaky and I think a lot of people would miss why the boundary case matters.

Questions Asked (2)

Given random variables X, Y, Z with corr(X,Y) = 0.8 and corr(X,Z) = 0.9, find the full range of possible values for corr(Y,Z). Use the fact that a valid correlation matrix must be positive semidefinite (nonneg determinant) to derive both bounds.

Technical Trade-offsData Modeling

Author's notes

This took me a minute to set up properly.

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AI HintsAI Generated

Suggested Approach

Set up the 3x3 correlation matrix with unknown ρ = corr(Y,Z) and compute its determinant. The positive semidefinite condition (determinant ≥ 0) yields a quadratic inequality in ρ, which you solve to find the lower and upper bounds. Verify that the bounds are attainable by checking that the matrix is PSD at the endpoints.

Pro tip: Mention that the bounds are tight and can be achieved by specific linear combinations of X and an independent noise term; this shows you understand the geometry of correlation matrices beyond just the algebra.

1. Write the correlation matrix

Construct the 3x3 matrix with 1s on the diagonal, 0.8 for corr(X,Y), 0.9 for corr(X,Z), and ρ for corr(Y,Z).

2. Compute the determinant

Calculate det(R) as a function of ρ. For a 3x3 correlation matrix, det(R) = 1 + 2*0.8*0.9*ρ - 0.8^2 - 0.9^2 - ρ^2.

3. Apply positive semidefinite condition

Set det(R) ≥ 0 and solve the resulting quadratic inequality: -ρ^2 + 1.44ρ - 0.45 ≥ 0, which simplifies to ρ^2 - 1.44ρ + 0.45 ≤ 0.

4. Solve for bounds

Find the roots of the quadratic: ρ = [1.44 ± sqrt(1.44^2 - 4*0.45)]/2 = [1.44 ± sqrt(2.0736 - 1.8)]/2 = [1.44 ± sqrt(0.2736)]/2 ≈ [1.44 ± 0.523]/2, giving ρ ≈ 0.4585 and ρ ≈ 0.9815. Thus the range is approximately [0.4585, 0.9815].

5. Verify attainability

Confirm that at the endpoints the determinant is zero, so the matrix is PSD (positive semidefinite) and the bounds are achievable.

Key Points to Mention

Correlation matrix must be symmetric and positive semidefinite.
For a 3x3 matrix, positive semidefiniteness is equivalent to nonnegative determinant and nonnegative principal minors.
The determinant condition gives a quadratic inequality in ρ.
The bounds are tight and can be achieved by constructing Y and Z as linear combinations of X and independent noise.
The result shows that high correlation with a common variable forces a minimum correlation between the other two.
Always check that the resulting matrix is valid (e.g., all eigenvalues nonnegative) at the bounds.

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

Can you construct three random variables where every pairwise correlation equals -1/2? If yes, give an explicit construction. If no, explain why that value is exactly the boundary of what's achievable.

Technical Trade-offsAlgorithms & Data Structures

Author's notes

The answer is yes, it's achievable, but barely.

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AI HintsAI Generated

Suggested Approach

First, recognize that the question is about the feasibility of a correlation matrix with all off-diagonal entries equal to -1/2. Use linear algebra to check positive semidefiniteness: the matrix with 1 on the diagonal and -1/2 off-diagonal has eigenvalues 0 and 1.5, so it is PSD and thus achievable. Then construct three random variables explicitly, for example by taking X1, X2, X3 as linear combinations of two independent standard normals with appropriate coefficients.

Pro tip: Mention that -1/2 is the exact lower bound for pairwise correlation among three variables because the sum of all pairwise correlations must be at least -1. This shows you understand the boundary condition and can generalize.

1. Formulate the correlation matrix

Write down the 3x3 matrix with 1 on the diagonal and -1/2 on all off-diagonal entries. This matrix must be positive semidefinite for the correlations to be realizable.

2. Check positive semidefiniteness

Compute the eigenvalues of the matrix. The eigenvalues are 0 (with multiplicity 1) and 1.5 (with multiplicity 2), so the matrix is PSD. Therefore, such random variables exist.

3. Construct explicit random variables

Let Z1 and Z2 be independent standard normal variables. Define X1 = Z1, X2 = -1/2 Z1 + (√3/2) Z2, and X3 = -1/2 Z1 - (√3/2) Z2. Verify that each has variance 1 and pairwise correlations are -1/2.

4. Explain the boundary

Show that for any three random variables with unit variance, the sum of all pairwise correlations is at least -1. This implies each pairwise correlation cannot be less than -1/2. Since we achieved -1/2, it is the exact boundary.

Key Points to Mention

Correlation matrix must be positive semidefinite.
Eigenvalues of the matrix with 1 on diagonal and -1/2 off-diagonal are 0 and 1.5.
Explicit construction using independent standard normals.
Proof that the sum of pairwise correlations is at least -1.
Generalization: for n variables, the minimum common pairwise correlation is -1/(n-1).
Connection to linear algebra and geometry of random variables.

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