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HMMT 二月 2003 · CALC 赛 · 第 1 题

HMMT February 2003 — CALC Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  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?
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 center is 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 is π π ∫ 1 2 r cot 2 r dr = . 0 3