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

图老师2

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

图老师2

图老师2

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.