Three points are chosen independently and uniformly inside the unit square in the plane. Find the expected area of the smallest closed rectangle that has sides parallel to the coordinate axes and that contains the three points.
2026-04-02 06:48:02.1775112482
Expected area of the smallest closed rectangle
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It will be $\mathsf E\Big((\max\limits_{i=1}^3\{X_i\} - \min\limits_{i=1}^3\{X_i\})\cdot(\max\limits_{i=1}^3\{Y_i\} - \min\limits_{i=1}^3\{Y_i\})\Big)$ where $\big(\langle X_i,Y_i\rangle\big)_{i=1}^3\overset{iid}\sim\mathcal U(0;1)^2$.
Use the independence of the given random variables, the Law of Total Expectation, and what you know about order statistics for a sample of iid uniform distributions.