图老师2 发表于 2019-8-3 08:37

在半径为 1 的球面上随机取 P,Q 两点，求 P,Q 两点之间空间直线距离的期望值

本帖最后由 luyuanhong 于 2019-8-5 23:10 编辑

在半径为1 的球面上所有两点之间的直线距离的平均值是多少?

luyuanhong 发表于 2019-8-6 00:37



图老师2 发表于 2019-8-6 06:52

:lol:lol:lol

图老师2 发表于 2019-8-6 07:03

It is sufficient to fix one point and to find the average distance of the other points from it. Take the point as P and the centre as O. Now consider a general point Q on the surface. Let angle POQ = θ. The distance PQ is 2 sin θ/2 and this is the same for all points in a band angular width dθ at the angle θ. The band has radius sin θ. Hence the average distance is 1/4π ∫0π (2π sin θ) (2 sin θ/2) dθ = ∫ 4 sin2θ/2 d(sin θ/2) = 4/3.
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