Section01_契比雪夫不等式

$$\begin{split} \mathbb{P}\{\vert X-EX \vert \ge \epsilon\} \le \frac{DX}{\epsilon^{2}} \\ \mathbb{P}\{\vert X-EX \vert < \epsilon\} \ge 1 - \frac{DX}{\epsilon^{2}} \end{split}$$

• 例 设随机变量 $X_{1},X_{2},\cdots,X_{n}$ 独立同分布，且 $X_{1}$ 的四阶矩存在，记 $u_{k} = E(X_{1}^{k}), \quad k = 1,2,3,4$，由契比雪夫不等式，对 $\forall\ \epsilon>0$，有 $\mathbb{P}\{\vert \frac{1}{n}\sum\limits_{i=1}^{n}X_{i}^{2} - u_{2} \vert \ge \epsilon\} \le$ ( ) $$\begin{array}{ll} \because & X_{1},X_{2},\cdots,X_{n}\text{独立同分布} \\ \therefore & E(\frac{1}{n}\sum\limits_{i=1}^{n} X^{2}_{i}) = \frac{1}{n}\sum\limits_{i=1}^{\infty}EX_{i}^{2} = \frac{1}{n}\times n\times EX_{1}^{2} = u_{2}\\ & D(\frac{1}{n}\sum\limits_{i=1}^{n} X^{2}_{i}) = \frac{1}{n^{2}}\sum\limits_{i=1}^{n}DX^{2}_{i} = \frac{1}{n^{2}}\times n DX^{2}_{1} = \frac{EX_{1}^{4} - (EX^{2})^{2}}{n} \\ & =\frac{u_{4} - u_{2}^{2}}{n} \\ \because & \mathbb{P}\{\vert X - EX \vert\ge \epsilon\} \le \frac{DX}{\epsilon^{2}} \\ \therefore & \mathbb{P}\{\vert \frac{1}{n}\sum\limits_{i=1}^{n}X_{i}^{2} - u_{2} \vert \ge \epsilon\} \le \frac{u_{4} - u_{2}^{2}}{n\epsilon^{2}} \\ \end{array}$$