Joint Distribution of Extrema for Uniform Variates

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Continuous Joint Distributions.

Your Task
Let $X, Y \sim \text{i.i.d. Uniform}(0, 1)$. Define $U = \min(X, Y)$ and $V = \max(X, Y)$.
(1) Find the joint probability density function $f_{U, V}(u, v)$.
(2) Compute $\text{Cov}(U, V)$ and the correlation coefficient $\rho(U, V)$.

Constraints

• Samples are independent and identically distributed on their respective support.
• Boundary conditions are defined strictly within standard real intervals.