The average area of a random rectangle

Question

Two side lengths X and Y are each chosen independently and uniformly at random from [0, 1], forming a rectangle with area XY. What is the expected (average) area of the rectangle?

Accepted Answer

Because X and Y are independent, the expected value of the product is the product of the expected values: E[XY] = E[X]\, E[Y] . Each side length is uniform on [0, 1], with mean 	frac12, so E[XY] = 	frac12 \cdot 	frac12 = \frac14 . Takeaway. For independent factors, expectations multiply, so the mean area is 	frac12 \cdot 	frac12 = 	frac14. Note this differs from the median-area threshold: the distribution of XY is skewed toward small values.

The average area of a random rectangle

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