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135 lines
4.5 KiB
Markdown
135 lines
4.5 KiB
Markdown
# 六、随机变量的数字特征
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## 1. 数学期望
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**定义**：
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离散型：设$P\{X = x_k\} = p_k$，若$\sum_{k=1}^{\infty} x_k p_k$绝对收敛，则
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$$E(X) = \sum_{k=1}^{\infty} x_k p_k$$
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连续型：若$\int_{-\infty}^{\infty} xf(x)dx$绝对收敛，则
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$$E(X) = \int_{-\infty}^{\infty} xf(x)dx$$
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**随机变量函数的期望**：
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设$Y = g(X)$，g是连续函数
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- 离散型：$E(Y) = E[g(X)] = \sum_{k=1}^{\infty} g(x_k)p_k$
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- 连续型：$E(Y) = E[g(X)] = \int_{-\infty}^{\infty} g(x)f(x)dx$
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**数学期望性质**：
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1. 设C是常数，则$E(C) = C$
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2. 设X是随机变量，C是常数，则$E(X + C) = E(X) + C$
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3. 设X是随机变量，C是常数，则$E(CX) = CE(X)$
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4. 设X,Y是两个随机变量，则$E(X \pm Y) = E(X) \pm E(Y)$（可推广到任意有限个）
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5. 设X,Y是相互独立的随机变量，则$E(XY) = E(X)E(Y)$（可推广到任意有限个）
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---
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## 2. 方差与标准差
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**定义**：$D(X) = E\{[X - E(X)]^2\}$
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**计算公式**：$D(X) = E(X^2) - [E(X)]^2$
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**标准差**：$\sigma(X) = \sqrt{D(X)}$
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**方差的计算**：
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离散型：$D(X) = \sum_{k=1}^{\infty} [x_k - E(X)]^2 p_k$
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连续型：$D(X) = \int_{-\infty}^{\infty} [x - E(X)]^2 f(x)dx$
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**方差性质**：
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1. 设C是常数，则$D(C) = 0$
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2. 设X是随机变量，C是常数，则$D(CX) = C^2D(X)$，$D(X + C) = D(X)$
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3. 设X,Y是两个随机变量，则
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$$D(X \pm Y) = D(X) + D(Y) \pm 2Cov(X,Y)$$
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特别地，若X,Y相互独立，则$D(X \pm Y) = D(X) + D(Y)$
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4. $D(X) = 0$的充要条件是X以概率1取常数$E(X)$，即$P\{X = E(X)\} = 1$
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---
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## 3. 协方差
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**定义**：$Cov(X,Y) = E\{[X - E(X)][Y - E(Y)]\}$
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**计算公式**：$Cov(X,Y) = E(XY) - E(X)E(Y)$
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**性质**：
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1. $Cov(X,Y) = Cov(Y,X)$（对称性）
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2. $Cov(X,C) = 0$（C为常数）
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3. $Cov(X,X) = D(X)$
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4. $Cov(aX, bY) = ab \cdot Cov(X,Y)$，a,b是常数
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5. $Cov(X_1 + X_2, Y) = Cov(X_1, Y) + Cov(X_2, Y)$（双线性）
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6. 若X,Y相互独立，则$Cov(X,Y)=0$
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**与方差的关系**：
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$$D(X + Y) = D(X) + D(Y) + 2Cov(X,Y)$$
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---
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## 4. 相关系数
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**定义**：
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$$\rho_{XY} = \frac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}}$$
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**性质**：
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1. $|\rho_{XY}| \leq 1$
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2. $|\rho_{XY}| = 1$的充要条件是，存在常数a,b使$P\{Y = a + bX\} = 1$（线性关系）
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3. 若X,Y相互独立，则$\rho_{XY} = 0$（不相关）
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4. **不相关 ≠ 独立**：$\rho_{XY} = 0$只说明X,Y没有线性关系，可能有非线性关系
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**不相关的等价条件**（以下四条等价）：
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- $\rho_{XY} = 0$
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- $Cov(X,Y) = 0$
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- $E(XY) = E(X)E(Y)$
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- $D(X + Y) = D(X) + D(Y)$
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---
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## 5. 矩
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**定义**：设X和Y是随机变量
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| 矩的类型 | 定义 | 说明 |
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|---------|------|------|
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| **k阶原点矩** | $E(X^k)$，$k = 1,2,...$ | 一阶原点矩就是期望E(X) |
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| **k阶中心矩** | $E\{[X - E(X)]^k\}$，$k = 2,3,...$ | 二阶中心矩就是方差D(X) |
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| **k+l阶混合矩** | $E(X^k Y^l)$，$k,l = 1,2,...$ | |
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| **k+l阶混合中心矩** | $E\{[X-E(X)]^k[Y-E(Y)]^l\}$ | 二阶混合中心矩就是协方差Cov(X,Y) |
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---
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## 6. 切比雪夫不等式
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设$E(X)=\mu$，$D(X)=\sigma^2$存在，则对任意$\varepsilon>0$，
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$$P\{|X-\mu| \ge \varepsilon\} \le \frac{\sigma^2}{\varepsilon^2}$$
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等价地，
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$$P\{|X-\mu| < \varepsilon\} \ge 1 - \frac{\sigma^2}{\varepsilon^2}$$
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---
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## 7. 数字特征典型例题
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**例**：设随机变量$X \sim N(\mu, \sigma^2)$，$Y \sim N(\mu, \sigma^2)$，且设X,Y相互独立，求$Z_1 = \alpha X + \beta Y$和$Z_2 = \alpha X - \beta Y$的相关系数（其中$\alpha, \beta$是不为零的常数）。
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**解**：由于$X, Y \sim N(\mu, \sigma^2)$，可得
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$$E(X) = E(Y) = \mu, \quad D(X) = D(Y) = \sigma^2$$
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$Z_1$和$Z_2$的相关系数：
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$$\rho_{Z_1Z_2} = \frac{E(Z_1Z_2) - E(Z_1) \cdot E(Z_2)}{\sqrt{D(Z_1)} \cdot \sqrt{D(Z_2)}}$$
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由$E(Z_1) = E(\alpha X + \beta Y) = \alpha E(X) + \beta E(Y) = (\alpha + \beta)\mu$
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$E(Z_2) = E(\alpha X - \beta Y) = \alpha E(X) - \beta E(Y) = (\alpha - \beta)\mu$
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又$E(Z_1Z_2) = E[(\alpha X + \beta Y)(\alpha X - \beta Y)] = E(\alpha^2 X^2 - \beta^2 Y^2) = \alpha^2 E(X^2) - \beta^2 E(Y^2)$
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$= (\alpha^2 - \beta^2)(\sigma^2 + \mu^2)$
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$D(Z_1) = D(\alpha X + \beta Y) = \alpha^2 D(X) + \beta^2 D(Y) = (\alpha^2 + \beta^2)\sigma^2$
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$D(Z_2) = D(\alpha X - \beta Y) = \alpha^2 D(X) + \beta^2 D(Y) = (\alpha^2 + \beta^2)\sigma^2$
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于是
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$$\rho_{Z_1Z_2} = \frac{(\alpha^2 - \beta^2)(\sigma^2 + \mu^2) - (\alpha + \beta)\mu(\alpha - \beta)\mu}{\sqrt{(\alpha^2 + \beta^2)\sigma^2} \cdot \sqrt{(\alpha^2 + \beta^2)\sigma^2}} = \frac{(\alpha^2 - \beta^2)\sigma^2}{(\alpha^2 + \beta^2)\sigma^2} = \frac{\alpha^2 - \beta^2}{\alpha^2 + \beta^2}$$
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