Cube - Expected R^2

Calculus Level 3

Consider a cube of side length 1, centered on the origin. Suppose we think of the cube as a collection of points in spherical coordinates, with each point having a radius R R with respect to the origin, as well as angles θ \theta and ϕ \phi .

If points are chosen randomly and uniformly over the cube's surface, what is the expected value of R 2 R^2 ?


The answer is 0.41667.

Steven Chase by Steven Chase , 37, USA

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1 solution

Brian Moehring
Aug 27, 2018

Even though it looks harder than the previous problem , we can directly extend my solution there to see that R 2 R^2 has the same distribution as X 2 + Y 2 + ( 1 2 ) 2 X^2 + Y^2 + \left(\frac{1}{2}\right)^2 where X , Y X,Y are independent and uniformly distributed on [ − 1 2 , 1 2 ] . \left[-\frac{1}{2},\frac{1}{2}\right].

It follows that E [ R 2 ] = 1 4 + 2 E [ X 2 ] = 1 4 + 2 ( 1 12 ) = 5 12 ≈ 0.41666667 \mathbb{E}[R^2] = \frac{1}{4} + 2\mathbb{E}[X^2] = \frac{1}{4} + 2\left(\frac{1}{12}\right) = \frac{5}{12} \approx \boxed{0.41666667}

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