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空盒数的期望与方差

Find the expectation and the variance of the number of empty boxes.

专题
Probability / 概率
难度
L4

题目详情

A total of NN balls is placed into NN boxes in such a way that each ball is equally likely to be placed in each of the boxes and the placements are independent of each other. Find the expectation and the variance of the number of empty boxes.

解析

令 Xi=1{第 i 个盒子为空}X_i=\mathbf{1}\{\text{第 }i\text{ 个盒子为空}\}，空盒总数 N∅=∑i=1NXiN_\emptyset=\sum_{i=1}^N X_i。

单个盒子为空：

p=P(X1=1)=(N−1N)N.p=\mathbb{P}(X_1=1)=\left(\frac{N-1}{N}\right)^N.

所以

E[N∅]=Np=N(N−1N)N.\boxed{\mathbb{E}[N_\emptyset]=Np=N\left(\frac{N-1}{N}\right)^N}.

再令

q=P(X1=X2=1)=(N−2N)N.q=\mathbb{P}(X_1=X_2=1)=\left(\frac{N-2}{N}\right)^N.

则

Var⁡(N∅)=Np(1−p)+N(N−1)(q−p2).\boxed{\operatorname{Var}(N_\emptyset)=Np(1-p)+N(N-1)(q-p^2)}.