Correlation Structure, Regression Slopes, Covariance of Order Statistics, and Change-of-Variables

You are given standard random variables and asked to assess correlation constraints, regression relationships, a covariance involving order statistics, and a change-of-variables result. Assume variables have finite second moments and differentiability where needed.

Answer all parts:

1. Three random variables X, Y, Z have 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 from regressing Y on X. Derive the slope coefficient from regressing X on Y.
3. Let X and Y be i.i.d. Uniform(0, 1). Compute the covariance between the maximum and minimum of X and Y.
4. Suppose Y = g(X), where g is monotone (strictly increasing or decreasing) and differentiable. Given the pdf of Y, derive the pdf of X (the inverse-function distribution result).