HMMT 二月 2003 · CALC 赛 · 第 1 题

英文解析

1. A point is chosen randomly with uniform distribution in the interior of a circle of radius
2. What is its expected distance from the center of the circle?
   Solution: 2 / 3
   The probability of the point falling between a distance r and r + dr from the centeris the ratio of the area of the corresponding annulus to the area of the whole circle:
   2 2
   π [( r + dr ) − r ]
   2 πr dr
   → = 2 r dr for small values of dr . Then the expected distance isπ π
   ∫
   r cot 2 r dr = .21
   30