Abstract: 本文介绍多项式分布的相关知识
Keywords: The Multinomial Distributions

# 多项式分布

## 多项式分布的定义和导出 Definition and Derivation of Multinomial Distribution

$$f(\vec{x}|4,\vec{p})=Pr(X_A=x_1,X_B=x_2,X_o=x_3,X_{AB}=x_4)\\ =\begin{cases} \begin{pmatrix} &n&\\ x_1&x_2&x_3&x_4 \end{pmatrix}p_A^{x_1}p_B^{x_2}p_o^{x_3}p_{AB}^{x_4}&\text{if } x_1+x_2+x_3+x_4=n\\ 0&\text{otherwise} \end{cases}$$

$$f(\vec{x}|n,\vec{p})= \begin{cases} \begin{pmatrix} &n&\\ x_1&\dots&x_k \end{pmatrix}p_1^{x_1}\dots p_{k}^{x_k}&\text{if } x_1+\dots+x_k=n\\ 0&\text{otherwise} \end{cases}\tag{5.9.1}$$

Definition Multinomial Distributions.A discrete random vector $\vec{X}=(X_1,\dots,X_k)$ whose p.f. is given Eq(5.9.1) has the multinomial distribution with parameters $n$ and $\vec{p}=(p_1,\dots,p_k)$ .

## 多项式分布和二项分布的关系 Relation between the Multinomial and Binomial Distributions

Theorem Suppose that the random vector $\vec{X}=(X_1,X_2)$ has the multinomial distribution with parameters $n$ and $\vec{p}=(p_1,p_2)$ .Then $X_1$ has the binomial distribution with parameters $n$ and $p_1$ ,and $X_2=n-X_1$

Corollary Suppose that the random vector $\vec{X}=(X_1,\dots,X_k)$ has the multinomial distribution with parameters $n$ and $\vec{p}=(p_1,\dots,p_k)$ .The marginal distribution of each variable $X_i(i=1,\dots,k)$ is the binomial distribution with parameters $n$ and $p$

$$f_3(x_3)=\sum_{\text{all }x_1}\sum_{\text{all }x_2}f(x_1,x_2,x_3)\text{ for }x_1+x_2+x_3=n$$

Corollary Suppose that the random vector $\vec{X}=(X_1,\dots,X_k)$ has the multinomial distribution with parameters $n$ and $\vec{p}=(p_1,\dots,p_k)$ with $k > 2$ .Let $\ell<k$ ,and let $i_1,\dots,i_{\ell}$ be distinct elements of the set ${1,\dots,k}$ .The distribution of $Y=X_{i_1}+\dots+X_{i_{\ell}}$ is the binomial distribution with parameters $n$ and $p_{i_1}+\dots+p_{i_{\ell}}$

## 均值，方差，协方差 Means,Variances and Covariances

Theorem Means,Variances,and Covariances.Let the random vector $X$ have the multinomial distribution with parameters $n$ and $p$ .The means and variances of the coordinates of $X$ are
$$E(X_i)=np_i\text{ and } Var(X_i)=np_i(1-p_i)\text{ for }i=1,\dots,k$$
the covariances between the coordinates are
$$Cov(X_i,X_j)=-np_ip_j$$

$X_i+X_j$ 为一个随机变量，其他随机变量相加为另一个随机变量，那么新的分布是一个二项分布$p=p_i+p_j$ 以及$n=n$ 那么其分布是:
$$Var(X_i+X_j)=n(p_i+p_j)(1-p_i-p_j)$$

$$Var(X_i+X_j)=Var(X_i)+Var(X_j)-Cov(X_i,X_j)\\ =np_i(1-p_i)+np_j(1-p_j)-Cov(X_i,X_j)$$

\begin{aligned} n(p_i+p_j)(1-p_i-p_j)&=np_i(1-p_i)+np_j(1-p_j)-Cov(X_i,X_j)\\ Cov(X_i,X_j)&=-np_ip_j \end{aligned}

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