切比雪夫不等式 ≥ε≤

切比雪夫不等式

期望计算:
$$E(X)={\int }_{-\infty }^{+\infty }xf(x)dx$$
方差(方差顾名思义:（与均值的）差的平方)计算:
$$D(X)={\int }_{-\infty }^{+\infty }(x-E(X){)}^{2}f(x)dx$$
$$D(X)=E^2(X)-E(X^2)$$
切比雪夫不等式:
$$P(|x-E(x)|\ge \epsilon )\le \frac{D(x)}{{\epsilon }^2}$$

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$$\begin{gathered} D(X)={\int }_{-\infty }^{+\infty }(x-E(X){)}^{2}f(x)dx\ge \\ {\int }_{|x-E(X)|\ge \epsilon }(x-E(X){)}^{2}f(x)dx\ge \\ 在积分区间有|x-E(X)|\ge \epsilon \\ {\int }_{|x-E(X)|\ge \epsilon }{\epsilon }^{2}f(x)dx \end{gathered}$$
此切比雪夫不等式的证明参考