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  1. 038X and Y are independent random variables with the same variance. What is the correlation between X and X + Y?Statistics and estimationWarm upSquarepoint CapitalMontreal · 2026

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    1 over root 2, about 0.71. The covariance of X with X + Y is Var(X) plus Cov(X, Y), which is sigma squared plus zero. The standard deviation of X + Y is root 2 times sigma because the variances add. So the correlation is sigma squared over (sigma x root 2 sigma), which is 1/root 2. X explains half the variance of the sum, and the correlation is the square root of that half.

    What is the fastest way to set it up?

    A two-member team's score is the sum of both players' scores. If the players are equally good and play independently, knowing one player's score tells you something about the team total, but only half the story. Split the covariance: Cov(X, X + Y) = Cov(X, X) + Cov(X, Y) = sigma squared + 0. The variance of the sum is sigma squared + sigma squared = 2 sigma squared, because independent variances add. Correlation is covariance over the product of standard deviations: sigma squared over (sigma x root 2 sigma) = 1/root 2.

    X is half of X + Y: it explains half the variance, so rho = root(1/2)X, length sigmaYX + Yroot 2 sigma45 degreescos 45 = 0.707Variance of X + Y = 2 sigma squaredfrom X: sigma squaredfrom Y: sigma squaredCov(X, X + Y) = Var X + Cov(X, Y) = sigma squaredsd(X) x sd(X + Y) = sigma x root 2 sigmarho = sigma squared / (root 2 sigma squared)= 1 / root 20.707R squared = 0.5: X explains half of the sum
    Drawn as arrows, independent X and Y sit at right angles and their sum lies at 45 degrees to X, so the correlation is cos 45, about 0.707; equivalently, X supplies half of the variance of X + Y, and the correlation is the square root of one half.

    Why is the answer not 0.5?

    Because 0.5 is the R squaredThe share of one variable variance explained by another; for a simple regression it is the correlation squared., not the correlation. X explains exactly half of the variance of X + Y, and correlation is the square root of the share of variance explained, so it is root 0.5, about 0.707. The geometric picture makes it stick: treat independent variables as arrows at right angles, and correlation as the cosine of the angle between arrows. X + Y sits at 45 degrees to X, and cos 45 is 0.707.

    Give the general version to show you own it. If Y has variance k times X's, the correlation is 1/root(1 + k): the more noise you add, the lower it falls. That is the logic behind a noisy signal: a forecast that is half signal and half independent noise, by variance, correlates about 0.71 with the signal, not 0.5.

    Where candidates lose it

    The common loss is answering 0.5 because X is half of the sum. That is the share of variance, and correlation is its square root.

    The other loss is saying zero because X and Y are independent. The sum contains X, so it cannot be independent of X. Split the covariance in one line and the answer falls out.

    What the interviewer asks next

    • What is the correlation between X + Y and X - Y?
    • Y has four times the variance of X. What is corr(X, X + Y) now?
    • What is the correlation between the sum of the first 10 and the sum of the first 20 of a series of independent returns?

    Asked at Squarepoint Capital, Desk Quant Analyst Interview, Montreal, 2026 (Wall Street Oasis): There were also 3-4 basic math/stats questions about mean, covariance, correlation, etc.