Expected Euclidean Distance Between Two Random Points in a Unit Square

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Multi-dimensional Integration / Geometric Probability.

Your Task
Let $P_1 = (X_1, Y_1)$ and $P_2 = (X_2, Y_2)$ be two independent points chosen uniformly at random inside the unit square $[0, 1]^2$. Derive the exact closed-form value of the expected Euclidean distance $\mathbb{E}[D] = \mathbb{E}\left[\sqrt{(X_1 - X_2)^2 + (Y_1 - Y_2)^2}\right]$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.