两路公交：平均等车时间

What is the average waiting time to get on a bus at that bus stop?

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

At a bus stop you can take bus #1 or bus #2. Bus #1 passes 10 minutes after bus #2 has passed, whereas bus #2 passes 20 minutes after bus #1 has passed. What is the average waiting time to get on a bus at that bus stop?

解析

两车构成 30 分钟周期：在每个周期内，先来 #1（距离上一班 #2 已过 10 分钟），再过 20 分钟来 #2。

设到站时刻 $X\sim U(0,30)$ （相对某班 #2 的到达）。若 $X\le 10$ ，等到 #1 需 $10-X$ ；若 $X>10$ ，等到 #2 需 $30-X$ 。

因此

\mathbb{E}[W]=\frac{1}{30}\left(\int_0^{10}(10-x)dx+\int_{10}^{30}(30-x)dx\right)=\boxed{\frac{25}{3}}\text{ 分钟}.