Solution
Correlation of the Extremes
Show the problem again
X and Y are independent standard normal random variables. Let M = max(X, Y) and m = min(X, Y). What is the correlation coefficient between M and m? Give an exact closed form, or a decimal rounded to 4 places.
Worked solution
The answer is 1/(π − 1) ≈ 0.4669. Two symmetric-function facts do all the work: M·m = XY and {M², m²} = {X², Y²}. Means: E[M] = 1/√π (since M = (X + Y)/2 + |X − Y|/2 and E|X − Y| = 2/√π), and E[m] = −1/√π. Covariance: E[Mm] = E[XY] = 0, so Cov(M, m) = 0 − (1/√π)(−1/√π) = 1/π. Variances: E[M²] + E[m²] = E[X²] + E[Y²] = 2, and E[M²] = E[m²] because negating both variables swaps M and −m. So E[M²] = 1 and Var(M) = Var(m) = 1 − 1/π. Hence corr(M, m) = (1/π)/(1 − 1/π) = 1/(π − 1). Sorting two independent variables manufactures a substantial positive correlation.
Source: Classic Gaussian exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.