Citadel · Probability & Brainteasers

GQS Quant Phone — Correlation Range + Box-Muller-Style Sampling

TrueInterview

July 18, 2026 · 1 min read

Requirements

The phone screen lasts 45–60 minutes and begins with an intensive review of the resume, followed by brief verbal questions. Two questions were reported:

Let X, Y, and Z be random variables with corr(X, Y) = 0.8 and corr(X, Z) = 0.9. What values can corr(Y, Z) take? Follow-up: is it possible to construct X, Y, and Z for which corr(X, Y) = corr(Y, Z) = corr(X, Z) = -1/2?
Given access to an RNG producing samples from Uniform(0, 1), how would you generate a sample from Normal(0, 1)?

The interview reportedly begins with a behavior-oriented question concerning previous research and motivation, then ends with another one or two behavioral questions that carry unusually large weight in the final decision.

Notes

Correlation bound argument: The correlation matrix $\Sigma$ for (X, Y, Z) has to be positive semi-definite. Writing corr(X, Y) = a, corr(X, Z) = b, and corr(Y, Z) = c, the corresponding 3x3 correlation matrix is PSD exactly when its determinant satisfies 1 + 2abc - a^2 - b^2 - c^2 >= 0. Treating this as a quadratic in c and substituting a = 0.8 and b = 0.9 gives ab - sqrt((1-a^2)(1-b^2)) <= c <= ab + sqrt((1-a^2)(1-b^2)) = 0.72 ± sqrt(0.36 * 0.19) = 0.72 ± sqrt(0.0684) ≈ 0.72 ± 0.2615. Therefore, approximately, corr(Y, Z) ∈ [0.4585, 0.9815].
All-equal negative correlations: Substituting $a = b = c = \rho$ into the same PSD requirement produces $1 + 2\rho^3 - 3\rho^2 >= 0$ . At $\rho = -1/2$ , the expression becomes 1 - 1/4 - 3/4 = 0, meaning the matrix is rank-deficient while remaining PSD, so such a construction is possible. For an explicit realization, use three unit vectors in two dimensions separated by 120° like an equilateral triangle; each pair has inner product cos(120°) = -1/2.
Uniform to Normal: Two usual approaches are available. With Box-Muller, take independent U1, U2 ~ Uniform(0, 1) and define Z1 = sqrt(-2 ln U1) cos(2π U2) and Z2 = sqrt(-2 ln U1) sin(2π U2); the resulting variables are independent standard normals. Another option is the inverse-CDF approach, applying the standard-normal quantile function $\Phi^{-1}(U)$ , evaluated with a rational approximation such as Beasley-Springer-Moro.
Candidates reportedly underestimate how heavily the interviewer's closing behavioral questions influence the outcome. Prepare the

Requirements

The phone screen lasts 45–60 minutes and begins with an intensive review of the resume, followed by brief verbal questions. Two questions were reported:

Let X, Y, and Z be random variables with corr(X, Y) = 0.8 and corr(X, Z) = 0.9. What values can corr(Y, Z) take? Follow-up: is it possible to construct X, Y, and Z for which corr(X, Y) = corr(Y, Z) = corr(X, Z) = -1/2?

Given access to an RNG producing samples from Uniform(0, 1), how would you generate a sample from Normal(0, 1)?

Notes

Correlation bound argument: The correlation matrix

\Sigma

for (X, Y, Z) has to be positive semi-definite. Writing corr(X, Y) = a, corr(X, Z) = b, and corr(Y, Z) = c, the corresponding 3x3 correlation matrix is PSD exactly when its determinant satisfies 1 + 2abc - a^2 - b^2 - c^2 >= 0. Treating this as a quadratic in c and substituting a = 0.8 and b = 0.9 gives

ab - sqrt((1-a^2)(1-b^2)) <= c <= ab + sqrt((1-a^2)(1-b^2)) = 0.72 ± sqrt(0.36 * 0.19) = 0.72 ± sqrt(0.0684) ≈ 0.72 ± 0.2615

. Therefore, approximately, corr(Y, Z) ∈ [0.4585, 0.9815].

All-equal negative correlations: Substituting

a = b = c = \rho

into the same PSD requirement produces

1 + 2\rho^3 - 3\rho^2 >= 0

. At

\rho = -1/2

, the expression becomes 1 - 1/4 - 3/4 = 0, meaning the matrix is rank-deficient while remaining PSD, so such a construction is possible. For an explicit realization, use three unit vectors in two dimensions separated by 120° like an equilateral triangle; each pair has inner product cos(120°) = -1/2.

Uniform to Normal: Two usual approaches are available. With Box-Muller, take independent U1, U2 ~ Uniform(0, 1) and define Z1 = sqrt(-2 ln U1) cos(2π U2) and Z2 = sqrt(-2 ln U1) sin(2π U2); the resulting variables are independent standard normals. Another option is the inverse-CDF approach, applying the standard-normal quantile function

\Phi^{-1}(U)

, evaluated with a rational approximation such as Beasley-Springer-Moro.

Candidates reportedly underestimate how heavily the interviewer's closing behavioral questions influence the outcome. Prepare the