# 馬爾可夫不等式

## 表达式

X为一非负随机变量，则

${\displaystyle \mathrm {P} (X\geq a)\leq {\frac {\mathrm {E} (X)}{a}}.}$[1]

${\displaystyle \mu (\{x\in X:|f(x)|\geq \epsilon \})\leq {1 \over \epsilon }\int _{X}|f|\,d\mu .}$

### 对于单调增加函数的扩展版本

φ是定义在非负实数上的单调增加函数，且其值非负，X是一个随机变量，a ≥ 0，且φ(a) > 0，则

${\displaystyle \mathbb {P} (|X|\geq a)\leq {\frac {\mathbb {E} (\varphi (|X|))}{\varphi (a)}}}$

## 证明

{\displaystyle {\begin{aligned}{\textrm {E}}(X)&=\int _{-\infty }^{\infty }xf(x)dx\\&=\int _{0}^{\infty }xf(x)dx\\[6pt]&\geqslant \int _{a}^{\infty }xf(x)dx\\[6pt]&\geqslant \int _{a}^{\infty }af(x)dx\\[6pt]&=a\int _{a}^{\infty }f(x)dx\\[6pt]&=a{\textrm {P}}(X\geqslant a).\end{aligned}}}

## 用來推导柴比雪夫不等式

${\displaystyle \Pr(|X-{\textrm {E}}(X)|\geq a)\leq {\frac {{\textrm {Var}}(X)}{a^{2}}},}$

${\displaystyle \operatorname {Var} (X)=\operatorname {E} [(X-\operatorname {E} (X))^{2}].}$

${\displaystyle (X-\operatorname {E} (X))^{2}}$

${\displaystyle \Pr((X-\operatorname {E} (X))^{2}\geq a^{2})\leq {\frac {\operatorname {Var} (X)}{a^{2}}},}$

## 矩陣形式的馬可夫不等式

${\displaystyle M\succeq 0}$為自共軛矩陣形式的隨機變數，且${\displaystyle a>0}$，則

${\displaystyle \Pr(M\npreceq a\cdot I)\leq {\frac {\mathrm {tr} \left(E(M)\right)}{a}}.}$

## 應用實例

• 馬爾可夫不等式可用來證明切比雪夫不等式
• 馬爾可夫不等式可用來證明一個非負的隨機變數，其平均值${\displaystyle \mu }$和中位數${\displaystyle m}$滿足${\displaystyle m\leq 2\mu }$的關係。

## 參考資料

1. ^ Sheldon M Ross. Introduction to probability and statistics for engineers and scientists. Academic Press. 2009: 第127頁. ISBN 9780123704832.
2. ^ E.M. Stein, R. Shakarchi, "Real Analysis, Measure Theory, Integration, & Hilbert Spaces", vol. 3, 1st ed., 2005, p.91
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