Abstract: 本文主要介绍超几何分布
Keywords: Hypergeomtirc Distribution,Finite Population Correction

# 超几何分布

## 超几何分布定义和例子 Definition and Examples

$$Pr(X_2=1|X_1=0)=\frac{A}{A+B-1}>Pr(X_2=1|X_1=1)=\frac{A-1}{A+B-1}$$

Theorem Probability Function .The distribuiton of $X$ is Example has the p.f.
$$f(x|n,A,B)= \begin{cases} \frac{ \begin{pmatrix}A\\x\end{pmatrix} \begin{pmatrix}B\\n-x\end{pmatrix} }{ \begin{pmatrix}A+B\\n\end{pmatrix} }&\text{for }max(0,n-B)\leq x\leq min(n,A)\\ 0&\text{otherwise} \end{cases}$$

Definition Hypergeomtric Distribution.Let $A,B$ and $n$ be nonnegative integers with $n\leq A+B$ .If a random variable $X$ has a discrete distribution with p.f. as in upside,then it is said that $X$ has the hypergeometric distribution with parameters $A,B$ and $n$

## 超几何分布的均值和方差 The Mean and Variance for a Hypergeomtirc Distribution

Theorem Mean and Variance.Let X have a hypergeometric distribution with strictly positive parameters A,B and n.Then:
$$E(X)=\frac{nA}{A+B}\\ Var(X)=\frac{nAB}{(A+B)^2}\times \frac{A+B-n}{A+B-1}$$

$$R_1,B_1,B_2\\ R_1,B_2,B_1\\ B_1,R_1,B_2\\ B_1,B_2,R_1\\ B_2,R_1,B_1\\ B_2,B_1,R_1$$

1. 把全部的取球过程列举出来，避免加权，我们把所有球都当做不同球。
2. 当把所有的结果列举出来，每行之间是互斥的。把前面每步取球相互影响的试验变成了互不影响的试验

$$Var(X_i)=\frac{AB}{(A+B)^2}$$

$$Var(X)=\sum^{n}_{i=1}Var(X_i)+2{\sum\sum}_{i<j}Cov(X_i,X_j)$$

\begin{aligned} Cov(X_1,X_2)&=-\frac{AB}{(A+B)^2(A+B-1)}\\ Var(X_i,X_j)&=\frac{nAB}{(A+B)^2}-\frac{n(n-1)AB}{(A+B)^2(A+B-1)}\\ &=\frac{nAB(A+B-1)-n(n-1)AB}{(A+B)^2(A+B-1)}\\ &=\frac{nAB}{(A+B)^2}\frac{A+B-n}{(A+B-1)} \end{aligned}

## 抽样方法比较 Comparison of Sampling Methods

Theorem $a_n$ and $c_n$ be sequences of real numbers such that $a_n$ converges to $0$ ,and $c_na_n^2$ converges to $0$ .Then
$$lim_{n\to \infty}(1-a_n)^{c_n}e^{-a_nc_n}=1$$
In particular,if $a_nc_n$ converges to $b$ ,then $(1+a_n)^{c_n}$ converges to $e^b$

Theorem Closeness of Binomial and Hypergeometric Distribution .Let $0< p < 1$ ,and let $n$ be a positive integer.Let $Y$ have the binomial distribution with parameters $n$ and $p$ .For each positive integer $T$ ,let $A_T$ and $B_T$ be integers such that $lim_{T\to \infty}A_{T}=\infty$ , $lim_{T\to \infty}B_{T}=\infty$ ,and $lim_{T\to \infty} A_T/(A_T+B_T)=p$ .Let $X_T$ have the hypergeometric distribution with parameters $A_T,B_T$ ,and $n$ .For each fixed $n$ and each $x=0,\dots,n$
$$lim_{T\to \infty}\frac{Pr(Y=x)}{Pr(X_T=x)}=1$$

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