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Derive Coefficient and Covariance in Regression Analysis

CitadelData Scientist · Technical Screen
MediumData ScientistTechnical ScreenStatistics & Math

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Evaluates statistics fundamentals across equicorrelation constraints, reverse regression slopes, covariance of uniform order statistics, and change of variables. Strong answers use positive semidefinite matrices, simple-regression identities, order-statistic expectations, and Jacobian transformations.

Scenario

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

Question

  1. For three random variables X, Y, Z with identical pairwise correlations ρ, what is the smallest possible value of ρ?

  2. In simple linear regression of Y on X, you know R² and the slope coefficient β(y|x). Derive the slope β(x|y) from regressing X on Y.

  3. Let X and Y be i.i.d. Uniform(0, 1). Compute Cov(max(X, Y), min(X, Y)).

  4. Given a monotone function Y = g(X), derive the pdf of X from the pdf of Y (inverse-function distribution).

Hints

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