断棍：较短段长度的期望

A stick of length 1

A stick of length 1 drops and breaks at a random place uniformly distributed across the length. What is the expected length of the smaller part?

解析

断点 $X\sim U(0,1)$ ，两段长度为 $X$ 与 $1-X$ ，较短段 $L=\min(X,1-X)$ 。

\mathbb{E}[L]=\int_0^{1/2}x\,dx+\int_{1/2}^{1}(1-x)\,dx=\frac18+\frac18=\boxed{\frac14}.