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.
Expected area of the smallest closed rectangle
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probability
expectation
area
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0The area: $(\max (x_1,x_2,x_3) - \min(x_1,x_2,x_3))\times (\max (y_1,y_2,y_3) - \min(y_1,y_2,y_3))\\ E[\max (x_1,x_2,x_3)] = 0.75,E[\min (x_1,x_2,x_3)] = 0.25\\E[\max (x_1,x_2,x_3) - \min(x_1,x_2,x_3)] = E[\max (x_1,x_2,x_3) - E[\min(x_1,x_2,x_3)]$ – 2017-02-14
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0I don't see a cogent reason to close this question. – 2017-02-15
1 Answers
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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.
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0Dear Graham, I am interested in this problem, but i have no such knowledge to solve it. Can you explain it better or give me some support material? – 2017-02-18