MathProblem 17 Dartboard problem #1

A dart is thrown at a circular dart board of radius one. The dart can land at any place on the dartboard with equal probability. What is the mean distance between where the dart hits and the center of the board?

Solution

一个半径为 \(1\) 的圆盘，向上面投掷飞镖，问落点距离原点的期望是多少。

假设距离为 \(x\), 我们需要求出 \(E[x]\):

\[E[x] = \int xp(x) \]

考虑计算 \(p(x)\)，我们利用圆环 \(x\sim x+dx\):

\[p(x) = \frac{S_x}{S_{all}} = \frac{2\pi x\cdot dx}{\pi \cdot 1^2}=2x\cdot dx \]

所以积分为：

\[E[x] = \int_{0}^1 2x^2 dx = \frac{2}{3} \]