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随机插入排名：Paula 的名次分布

Random Ranks

专题
Algorithmic Programming / 算法编程
难度
L4

题目详情

Peter draws n=100\mathrm{n} = 100 independent realizations of a continuous rv and ranks them in increasing order from 1 to 100. Subsequently, Paula draws a single value from the same population and inserts this value into the rank order created earlier by Peter. For example, if her value is such that 50 of Peter's draws are smaller and 50 are larger, then the rank associated with her draw would be 51—that is, overall, her value would be the 51st in increasing order. Or, if her value is smaller than all 100 of Peter's, then the rank 1 would be associated with it.

a. Is it more likely that Paula's value will occupy rank 51 than rank 1? b. Derive for general n the probability that Paula's value will occupy rank k, where 1≤k≤n+11\leq k\leq n + 1

解析

因为 101 个独立连续样本的相对次序（全排列）等可能。

Paula 的样本在 1..101 的每一个名次上对称，概率相同。

因此

P(名次=k)=1n+1,k=1,2,…,n+1.\boxed{\mathbb{P}(\text{名次}=k)=\frac{1}{n+1}},\quad k=1,2,\ldots,n+1.

对 n=100n=100，每个名次概率都是 1/1011/101，所以名次 51 与名次 1 一样可能。