Derive Coefficient and Covariance in Regression Analysis

##### 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.