## 3 随机向量及其分布

2019-04-07 18:20 CST
2019-07-23 00:06 CST
CC BY-NC 4.0

## 3.1 二维随机变量及其分布函数

### 一、二维离散型随机变量

• $p_{ij} \geqslant 0$，
• $\sum\limits_{i = 1}^{+\infty}\sum\limits_{j=1}^{+\infty} p_{ij} = 1$

$$P(x_1 = k_1, x_2 = k_2) = \dfrac{n!}{k_1!k_2! (n-k_1-k_2)!} p_1^{k_1} p_2^{k_2} (1 - p_1 - p_2)^{n - k_1 - k_2}​$$

$$P(x_1 = N_1, x_2 = N_2) = \dfrac{C_{N_1}^{N_1} C_{N_2}^{N_2} C_{N - N_1 - N_2}^{n - N_1 - N_2}}{C_N^n}​$$

$$F(x, y) = \sum\limits_{x_i \leqslant x, y_j \leqslant y} p_{ij}$$

$$p_{ij} = F(x_i, y_j) - F(x_i, y_{j-1}) - F(x_{i-1}, y_j) + F(x_{i-1}, y_{j-1})$$

• 单调性
• $0 \leqslant F(x, y) \leqslant 1$
• 右连续性
• 对任意实数$x_1 \leqslant x_2$, $y_1 \leqslant y_2$，有$$F(x_2, y_2) - F(x_1, y_2) - F(x_2, y_1) + F(x_1, y_1) \geqslant 0$$

### 二、二维连续型随机向量

$$F(x, y) = \int_{-\infty}^x \int_{-\infty}^y p(u, v) \text{d}u \text{d}v,$$

• 非负性
• 二维积分为1
• 设$G​$为平面上任一区域，则$$P((X, Y) \in G) = \iint\limits_G p(x, y) \text{d}x \text{d}y​$$
• 若$p(x, y)$在点$(x_0, y_0)$处连续，则$$\dfrac{\partial^2 F(x, y)}{\partial x \partial y} \Bigr\vert_{(x_0, y_0)} = p(x_0, y_0)$$

$X$的边缘分布为

$$F_X(x) = \int_{-\infty}^x \int_{-\infty}^{+\infty} p(u, v) \text{d}u \text{d}v​$$

$$p(x, y) = \begin{cases} \dfrac{1}{(b-a)(d-c)}, &(x, y) \in [a, b] \times [c, d], \\ 0, &(x, y) \notin [a, b] \times [c, d].\end{cases}$$

$$p(x, y) = \dfrac{1}{2\pi \sigma_1 \sigma_2 \sqrt{1 - \rho^2}} \exp \left\lbrace -\dfrac{1}{2 (1-\rho^2)} \left[ \left( \dfrac{x-\mu_1}{\sigma_1} \right)^2 - 2\rho \left( \dfrac{x-\mu_1}{\sigma_1} \right) \left( \dfrac{y-\mu_2}{\sigma_2} \right) + \left( \dfrac{y-\mu_2}{\sigma_2} \right)^2 \right] \right\rbrace$$

$$p_X(x) = \dfrac{1}{\sqrt{2\pi}\sigma_1} \exp \left\lbrace \dfrac{(x - \mu_1)^2}{2\sigma_1^2} \right\rbrace​$$

$$p_Y(y) = \dfrac{1}{\sqrt{2\pi}\sigma_2} \exp \left\lbrace -\dfrac{(y - \mu_2)^2}{2\sigma_2^2} \right\rbrace$$

（此部分内容就是上面的扩展，略）

## 3.2 条件分布

### 一、离散型随机向量的条件概率分布

$$P(Y = y_j | X = x_i) = \dfrac{P(X = x_i, Y = y_i)}{P(X = x_i)} = \dfrac{p_{ij}}{p_{i\cdot}}$$

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

$$P(Y = y_j) = \sum\limits_{i = 1}^{+\infty} P(Y = y_j, X = x_i) P(X = x_i)$$

### 二、连续性随机向量的条件概率

$$F_{Y|X=x}(y) = P(Y \leqslant y | X = x) = \int_{-\infty}^{y} \dfrac{p(x, v)}{p_X(x)} \text{d}v​$$

$$p_{Y|X=x}(y) = \dfrac{p(x, y)}{p_X(x)}$$

$$p(x, y) = p_{Y|X=x}(y) p_X(x)$$

## 3.3 随机变量的独立性

$$P(x_1 = x_1, \dots, x_n = x_n) = P(x_1 = x_1) \cdots P(x_n = x_n)$$

$$P(Y = y_j | X = x_j) = \dfrac{p_{ij}}{p_{i \cdot}} = p_{\cdot j} = P(Y = y_j)​$$

$$P(X \leqslant x, Y \leqslant y) = P(X \leqslant x) P(Y \leqslant y)$$

$$p(x, y) = g_1(x) g_2(y)$$

## 3.4 二维随机向量函数的分布

### 二、二维连续型随即向量函数的分布

#### 两个函数的联合分布

1. 存在唯一的逆变换 $$\begin{cases}x = x(u, v), \\ y = y(u, v).\end{cases}$$
2. 存在连续的一阶偏导数，且Jacobi行列式 $$J(u, v) = \left\vert \begin{matrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{matrix} \right\vert \neq 0$$