## 联合分布函数

$$F(x,y) = P ((X \le x)\cap(Y \le y)) = P (X \le x, Y \le y )$$

### 离散型随机变量联合分布

$$P(X=x_i, Y=y_i) = p_{ij},i,j=1,2,….$$

### 连续型随机变量联合分布

$$F(x,y) = \int_{-\infty}^y\int_{-\infty}^xf(u,v)dudv$$

1. $$f(x,y) \ge 0$$

2. $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)dxdy = F(\infty,\infty)$$

3.设 $G$ 是 $xOy$ 平面上的区域，点 $(X,Y)$ 落在G内的概率为

$$P((X,Y)\in G) = \int\int f(x,y)dxdy$$

4.若 $f(x,y)$ 在点 $(x, y)$ 连续，则

$$\frac{\partial^2F(X,Y)}{\partial x \partial y} = f(x, y)$$

## 边缘分布函数

$$F_X(x) = P(X \le x) = P(X \le x,Y \lt \infty) = F(x, \infty)$$

$$F_X(x) = F(x,\infty)$$

### 离散型随机变量边缘分布

$$p_{i.} = \sum_{j=1}^{\infty} p_{ij} = P(X = x_i), i=1,2,3…..n$$

$$p_{.j} = \sum_{i=1}^{\infty} p_{ij} = P(Y = y_j), j=1,2,3…..n$$

### 连续型随机变量边缘分布

$$F_X(x) = F(x,\infty) = \int_{-\infty}^{x}(\int_{-\infty}^{\infty}f(x,y)dy)dx$$

$$f_X(x) = \int_{-\infty}^{\infty}f(x,y)dy$$

## 条件分布

### 离散型随机变量的条件分布

$$P(X = x_i|Y= y_j) = \frac{P(X = x_i, Y=y_j)}{P(Y=y_j)} = \frac{p_{ij}}{p_{.j}}, i = 1,2,3$$

### 连续型随机变量的条件分布

$$f_{X|Y}(x|y) = \frac{f(x,y)}{f_Y(y)}$$

$$\int_{-\infty}^x f_{X|Y}(x|y)dx = \int_{-\infty}^x \frac{f(x,y)}{f_Y(y)}dx$$

$F_{X|Y}(x|y) = P(X \le x| Y=y)$

## 相互独立的随机变量

$F(x,y) = F_X(x)F_Y(y)$

$f(x,y) = f_X(x)f_Y(y)$

$P(X = x_i, Y = y_j) = P(X=x_i)P(Y=y_j)$

## 二维随机变量的函数的分布

### $Z = X + Y$ 的分布

$$f_{X+Y}(z) = \int_{-\infty}^{\infty} f(z-y,y)dy$$

$$f_{X+Y}(z) = \int_{-\infty}^{\infty} f(x,z-x)dx$$

$f(x,y) = f_X(x)f_Y(y)$

$$f_{X+Y}(z) = \int_{-\infty}^{\infty} f_X(z-y)f_Y(y)dy$$

$$f_{X+Y}(z) = \int_{-\infty}^{\infty} f_X(x)f_Y(z-x)dx$$

### $Z=XY$ 和 $Z=Y/X$ 的分布

$$f_{Y/X}(z) = \int_{-\infty}^{\infty} |x|f(x,xz)dx$$

$$f_{XY}(z) = \int_{-\infty}^{\infty} \frac{1}{|x|}f(x,z/x)dx$$

$$f_{Y/X}(z) = \int_{-\infty}^{\infty} |x|f_X(x)f_Y(xz)dx$$

$$f_{XY}(z) = \int_{-\infty}^{\infty} \frac{1}{|x|}f_X(x)f_Y(z/x)dx$$

### $M = max(X,Y)$ 和 $N = min(X,Y)$ 的分布

$F_{max}(z) = P(M \le z) = P(X \le z, Y \le z) = P(X \le z)P(Y \le z)$

$F_{max}(z) = F_X(z)F_Y(z)$

$F_{min}(z) = P(N \le z) = 1 - P(N \gt z) = 1 - P(X > z)P(Y>z)$

$F_{min}(z) = 1 - (1 - F_X(z))(1 - F_Y(z))$

$M = max \lbrace X_1,X_2…,X_n \rbrace$ 及 $N = min\lbrace X_1,X_2…,X_n \rbrace$ 的分布函数分别为

$$F_{max}(z) = F_{X_1}(z)F_{X_2}(z)…F_{X_n}(z)$$

$$F_{min}(z) = 1 - (1 - F_{X_1}(z))(1 - F_{X_2}(z))…(1 - F_{X_n}(z))$$

$$F_{max}(z) = [F(z)]^n$$

$$F_{min}(z) = 1 - (1 - F(z))^n$$