期望

定义

$$ E(X)=\sum_{i=1}^{\infty} x_{i}p_{i} $$

运算性质

1. 【线性】$$E\left( \sum_{i=1}^{n} \lambda_{i}X_{i} +c \right) = \sum_{i=1}^{n} \lambda_{i}E(X_{i})+c$$

2. 【独立】若 $X_{1}, X_{2}, \dots, X_{n}$ 相互独立则 $$E\left[ \prod_{i=1}^{n} g_{i}(X_{i}) \right] = \prod_{i=1}^{n} E(g_{i}(X_{i}))$$

方差

定义

运算性质

1. 【非负】

   1. $D(X) \ge 0$
   2. $E(X^{2}) \ge E^{2}(X)$

2. $D(X)=E(X^{2})-E^{2}(X)$

3. $D(c)=0$

4. $D(aX+b)=a^{2}D(X)$

5. $D(X\pm Y)=D(X)+D(Y)\pm2\mathrm{Cov}(X,Y)$

6. 若 $X_{1}, X_{2}, \dots, X_{n}$ 相互独立则 $$D\left[ \sum_{i=1}^{n} g_{i}(X_{i}) \right]=\sum_{i=1}^{n} D[g_{i}(X_{i})]$$

7. 【正态】若 $X\sim N(0,\sigma^{2})$，则 $E(X^{k})=(k-1)\sigma^{k}$