Jane Street·Data Scientist·Technical Phone Screen·Intermediate

IntermediatePrefer not to say

May 2026

Summary

Jane Street Data Scientist interview with a heavy probability/statistics focus. The core problem was a multivariate correlation question that required both a proof and a construction, which is pretty typical for quant-adjacent roles there but still caught me off guard with how much rigor they expected.

Questions Asked (1)

Given seven random variables on the same probability space, each with mean 0 and variance 1, identically distributed, and with the same pairwise correlation rho for every pair, what is the minimum possible value of rho? You need to prove the lower bound is tight by showing it can't go lower AND constructing a valid joint distribution that actually achieves it.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

This wrecked me for the first few minutes.

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AI HintsAI Generated

Suggested Approach

Use the fact that the correlation matrix must be positive semidefinite. For n=7 with all pairwise correlations equal to ρ, the matrix is (1-ρ)I + ρJ. Its eigenvalues are 1-ρ (multiplicity 6) and 1+6ρ (multiplicity 1). For PSD, we need 1-ρ ≥ 0 and 1+6ρ ≥ 0, giving ρ ≥ -1/6. To show tightness, construct a joint distribution achieving ρ = -1/6, e.g., using a random vector that is equally likely to be one of two orthogonal vectors.

Pro tip: Emphasize that the lower bound comes from the positive semidefiniteness of the correlation matrix, and that achieving it requires a careful construction that satisfies all constraints. Mention that this is a classic result in multivariate statistics and that the bound depends on the number of variables.

1. Set up the correlation matrix

Write the 7x7 correlation matrix with 1 on the diagonal and ρ on all off-diagonal entries. Recognize that any valid correlation matrix must be positive semidefinite (PSD).

2. Compute eigenvalues

Express the matrix as (1-ρ)I + ρJ, where J is the all-ones matrix. Its eigenvalues are 1-ρ (multiplicity 6) and 1+6ρ (multiplicity 1).

3. Derive the lower bound

For PSD, all eigenvalues must be nonnegative. Thus 1-ρ ≥ 0 and 1+6ρ ≥ 0. The first gives ρ ≤ 1, the second gives ρ ≥ -1/6. So the minimum possible ρ is -1/6.

4. Construct a distribution achieving ρ = -1/6

Let X be a random vector taking values in {±1}^7 with equal probability, but conditioned on the sum of coordinates being 0? Actually, we need a construction. One standard construction: Let Y be a random vector uniformly distributed on the vertices of a regular simplex? Alternatively, use the fact that the matrix with ρ = -1/6 is PSD, so there exists a multivariate normal distribution with that correlation matrix. Since the variables are identically distributed with mean 0 and variance 1, we can simply take a multivariate normal with mean 0 and covariance matrix having 1 on diagonal and -1/6 off-diagonal. This achieves the bound.

5. Verify the construction

Check that the covariance matrix is PSD (it is, by construction) and that the variables have mean 0, variance 1, and pairwise correlation -1/6. Thus the lower bound is tight.

Key Points to Mention

Positive semidefinite correlation matrix
Eigenvalues of equicorrelation matrix
Lower bound ρ ≥ -1/(n-1) for n variables
Construction using multivariate normal distribution
Tightness of the bound
Identically distributed and mean 0, variance 1 conditions

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