# 离散型随机变量的边缘分布

1，联合分布：我们以前知道二维随机变量 $$(X,Y)$$ 的联合分布函数为 $F(x,y)=P(X\le x,Y\le y)$

2，边缘分布函数：$F_X(x)=P(X\le x)=P(X\le x, Y<\infty)=F(x,+\infty)$

$F_Y(y)=P(Y\le y)=P(X< +\infty, Y\le y)=F(+\infty,y)$

3，离散型随机变量的边缘分布律：

$p_{i,\cdot}=P(X=x_i)=\sum_jP(X=x_i,Y=y_j)=\sum_jp_{ij}$

$p_{\cdot,j}=P(Y=y_j)=\sum_iP(X=x_i,Y=y_j)=\sum_ip_{ij}$

\begin{array}{|c|cccc|}\hline Y\Big{\backslash}X& 1&2&3&4\\ \hline 1&\frac{1}{16}& 0&0&0\\ 2&\frac{1}{16}&\frac{1}{12}&\frac{1}{8}&0\\ 3&\frac{1}{16}&\frac{1}{12}&\frac{1}{8}&0\\ 4&\frac{1}{16}&\frac{1}{12}&\frac{1}{8}&\frac{1}{4}\\ \hline\end{array}

\begin{array}{c|cccc}X&1&2&3&4\\ \hline p&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}\end{array}

$$Y$$ 的边缘分布律为：

\begin{align*}P(Y=1)&=P(X=1, Y=1)+P(X=2, Y=1)\\ &\quad +P(X=3, Y=1)+P(X=4, Y=1)\\ &=\frac{1}{16}+0+0+0=\frac{1}{16}\end{align*}

$P(Y=2)=\frac{7}{48},\quad P(Y=3)=\frac{13}{48},\quad P(Y=4)=\frac{25}{48}$

\begin{array}{c|cccc}Y&1&2&3&4\\ \hline p&\frac{1}{16}&\frac{7}{48}&\frac{13}{48}&\frac{25}{48}\end{array}

\begin{array}{|c|cccc|c|}\hline Y\Big{\backslash}X& 1&2&3&4&Y\\ \hline 1&\frac{1}{16}& 0&0&0&\frac{1}{16}\\ 2&\frac{1}{16}&\frac{1}{12}&\frac{1}{8}&0&\frac{7}{48}\\ 3&\frac{1}{16}&\frac{1}{12}&\frac{1}{8}&0&\frac{13}{48}\\ 4&\frac{1}{16}&\frac{1}{12}&\frac{1}{8}&\frac{1}{4}&\frac{25}{48}\\ \hline X&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&1\\ \hline\end{array}

\begin{array}{|c|cc|}\hline Y\Big{\backslash}X&1&2\\ \hline 1&\frac{1}{8}&\frac{1}{4}\\ 2&\frac{1}{8}&\frac{1}{2}\\ \hline\end{array}求 $$(X,Y)$$ 的边缘分布律。

\begin{array}{|c|cc|c|}\hline Y\Big{\backslash}X&1&2&Y\\ \hline 1&\frac{1}{8}&\frac{1}{4}&\frac{3}{8}\\ 2&\frac{1}{8}&\frac{1}{2}&\frac{5}{8}\\ \hline X&\frac{1}{4}&\frac{3}{4}&1\\ \hline\end{array}