What is the expected value of the distance you travel

markfang2050

You are randomly placed in the interior of a circle of radius 1 (uniform distribution). You randomly choose a direction (uniformly) and begin to walk that way, stopping when you get to the circle. What is the expected value of the distance you travel?

mathe

$8/{3\pi}$

mathe

首先各个方向是均匀的，不妨指定最后总是朝一个方向走，结果平均距离是相同的。不妨只计算向x轴正方向走的平均距离。
同样，总向正方形走和总向负方向走的平均距离也是相同的。对于园内任意一个点，上诉俩方法的总距离总是过这点水平方向上弦长。