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德国坦克/机车编号估计

The Locomotive Problem

专题
Statistics / 统计
难度
L4

题目详情

(a) A railroad numbers its locomotives in order, 1,2,…,N1, 2, \ldots , N . One day you see a locomotive and its number is 60. Guess how many locomotives the company has.

(b) You have looked at 5 locomotives and the largest number observed is 60. Again guess how many locomotives the company has.

解析

设公司共有 NN 台机车，编号 1,2,…,N1,2,\ldots,N，你观察到的样本来自“均匀不放回抽样”。若抽了 kk 台，最大编号为 MM，则

E[M]=k(N+1)k+1⇒N^=k+1kM−1\mathbb{E}[M]=\frac{k(N+1)}{k+1} \quad\Rightarrow\quad \widehat N=\frac{k+1}{k}M-1

是一个常用的无偏估计。

  • (a) k=1,M=60k=1,M=60：
N^=2⋅60−1=119.\boxed{\widehat N=2\cdot 60-1=119}.
  • (b) k=5,M=60k=5,M=60：
N^=65⋅60−1=71.\boxed{\widehat N=\frac{6}{5}\cdot 60-1=71}.

（补充：最大似然估计为 N^MLE=M=60\widehat N_{\mathrm{MLE}}=M=60。）