概率笔记 P04：随机变量的数字特征
1 数学期望
数学期望的定义
- 离散型，\(EX = \displaystyle \sum_{k = 1}^{+\infty} x_k p_k\)
- 连续型，\(EX = \displaystyle \int_{-\infty}^{+\infty} xf(x)dx\)
数学期望的性质
- \(E(C) = C\)
- \(E(aX+b) = aEX+b\)
- \(E(X \pm Y) = EX \pm EY\)
- \(X, Y\)相互独立，则\(E(XY) = EXEY\)
随机变量函数的期望
对于\(Y = g(X)\)，
- 离散型，\(EY = E(g(X)) = \displaystyle \sum_{k = 1}^{+\infty} g(x_k) p_k\)
- 连续型，\(EY = E(g(X)) = \displaystyle \int_{-\infty}^{+\infty} g(x) f(x)dx\)
二维随机变量函数的期望
对于\(Z = g(X, Y)\)，
- 离散型，\(EZ = E(g(X, Y)) = \displaystyle \sum_{i = 1}^{+\infty} \sum_{j = 1}^{+\infty} g(x_i, y_j) p_{ij}\)
- 连续型，\(EZ = E(g(X, Y)) = \displaystyle \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} g(x, y) f(x, y) dxdy\)
2 方差
方差的定义
方差\(DX = E\{[X - E(X)]^2\}\)。
标准差或均方差\(\sigma(X) = \sqrt{DX}\)。
方差的计算
\(DX = E(X^2) - (EX)^2\)
方差的性质
- \(D(C) = 0\)
- \(D(aX + b) = a^2 DX\)
- \(D(X \pm Y) = DX + DY \pm 2\mathrm {cov}(X,Y)\)
- 若\(X,Y\)相互独立，则\(D(aX+bY) = a^2DX + b^2DY\)，\(D(XY) = DX \cdot DY+DX(EY)^2 + DY(EX)^2\)
3 常用随机变量的数学期望和方差
- 0-1 分布，\(EX = p, DX = p(1-p)\)
- 二项分布\(X \sim B(n, p)\)，\(EX = np, DX = np(1-p)\)
- 泊松分布\(X \sim P(\lambda)\)，\(EX = \lambda, DX = \lambda\)
- 几何分布\(X \sim Ge(p)\)，\(EX = \cfrac 1p, DX = \cfrac {1-p}{p^2}\)
- 超几何分布\(X \sim H(N,M,n)\)，\(EX = n \cfrac MN, DX = \cfrac {nM(N-M)(N-n)}{N^2(N-1)}\)
- 均匀分布\(X \sim U(a, b)\)，\(E(X) = \cfrac {a+b}2, D(X) = \cfrac {(b-a)^2}{12}\)
- 指数分布\(X \sim E(\lambda)\)，\(E(X) = \cfrac 1\lambda, D(X) = \cfrac 1{\lambda^2}\)
- 正态分布\(X \sim N(\mu, \sigma^2)\)，\(E(X) = \mu, D(X) = \sigma^2\)
- \(\chi^2(n)\)分布，\(EX = n, DX = 2n\)
4 矩
矩的定义
- \(k\)阶原点矩：\(E(x^k)\)
- \(k\)阶中心矩：\(E[(X - EX)^k]\)
- \(X,Y\)的\(k+l\)阶混合矩\(E(X^k Y^l)\)
- \(X,Y\)的\(k+l\)阶混合中心矩\(E[(X - EX)^k (Y - EY)^l]\)
5 协方差
协方差的定义
- 协方差\(\mathrm{cov}(X, Y) = EE[(X - EX) (Y - EY)] = E(XY) - EXEY\)
- 协方差矩阵
\(\boldsymbol C = \left( \begin{matrix} c_{11} & c_{12} & \cdots & c_{1n} \\ c_{21} & c_{22} & \cdots & c_{2n} \\ \vdots & \vdots & & \vdots \\ c_{n1} & c_{n2} & \cdots & c_{nn} \\ \end{matrix} \right)\)，其中\(c_{ij} = \mathrm{cov}(X_i,X_j), \ c_{ij} = c_{ji}\)。
协方差的性质
- \(\mathrm{cov}(X, Y) = E(XY) - EXEY\)
- \(D(X \pm Y) = DX + DY \pm 2\mathrm{cov}(X, Y)\)
- \(\mathrm{cov}(X, Y) = \mathrm{cov}(Y, X)\)
- \(\mathrm{cov}(X,X) = DX\)
- \(\mathrm{cov}(X,C) = 0\)
- \(\mathrm{cov}(aX, bY) = ab \mathrm{cov}(X, Y)\)
- \(\mathrm{cov}(X_1 + X_2, Y) = \mathrm{cov}(X_1, Y) + \mathrm{cov}(X_2, Y)\)
6 相关系数
相关系数的定义
相关系数\(\rho_{XY} = \cfrac {\mathrm{cov}(X, Y)}{\sqrt{DX} \sqrt{DY}}\)
相关系数的性质
- \(|\rho_{XY}| \le 1\)
- \(|\rho_{XY}| = 1 \iff\)存在不全为零的常数\(a, b\)使得\(P\{aX +bY = 1\} = 1\)
- 若\(\rho_{XY} = 0\)，则\(X, Y\)不相关。
- \(\rho_{XX} = 1\)
- 若随机变量\(X, Y\)相互独立，则\(X, Y\)必不相关。反之不成立。
- 对于二维正态随机变量，相互独立和不相关是等价的。
