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递减抽数的期望

Select numbers

专题
General / 综合
难度
L4

题目详情

Select numbers uniformly distributed between 0 and 1, one after the other, as long as they keep decreasing; i.e. stop selecting when you obtain a number that is greater than the previous one you selected.

解析

设总抽样次数为 NN。对 k≥2k\ge 2，{N>k}\{N>k\} 等价于 X1>⋯>XkX_1>\cdots>X_k（前 kk 个严格递减），其概率为 1/k!1/k!。

用尾和公式：

E[N]=∑k≥0P(N>k)=1+1+∑k=2∞1k!=∑k=0∞1k!=e.\mathbb{E}[N]=\sum_{k\ge 0}\mathbb{P}(N>k)=1+1+\sum_{k=2}^{\infty}\frac{1}{k!}=\sum_{k=0}^{\infty}\frac{1}{k!}=\boxed{e}.