Assessing knowledge of correlation structure, regression relationships and covariance calculations.

For three random variables X, Y, Z with identical pairwise correlations ρ, what is the smallest possible value of ρ? Given R² and the coefficient β(y|x) from regressing Y on X, derive the coefficient β(x|y) from regressing X on Y. Let X and Y be i.i.d. Uniform(0,1). Compute Cov(max(X, Y), min(X, Y)). Given a monotone function Y = g(X), derive the pdf of X from the pdf of Y (inverse-function distribution).

Use positive-definite covariance matrices, β relations with R², Cov identities, and change-of-variables theorem.

Citadel Data Scientist interview question: Derive Coefficient and Covariance in Regression Analysis. {"blocks": [{"key": "583b9207", "text": "Scenario", "type": "header-two", "depth": 0, "inlineStyleRanges": [], "entityRanges": [], "data": {...