Union_Bound-布尔不等式-Boole's_inequality

Last updated Dec 4, 2021 Edit Source

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• This website explained it well: The Union Bound and Extension

$$\begin{aligned} P(A \cup B) &=P(A)+P(B)-P(A \cap B) \\ & \leq P(A)+P(B) . \end{aligned}$$

同样, 对于三个随机变量也有相应的不等关系:

$$\begin{aligned} P(A \cup B \cup C) &=P((A \cup B) \cup C) \\ & \leq P(A \cup B)+P(C) \\ & \leq P(A)+P(B)+P(C) \end{aligned}$$

In general, using induction we prove the following:

The Union Bound

For any events $A_{1}, A_{2}, \ldots, A_{n}$, we have $$ P\left(\bigcup_{i=1}^{n} A_{i}\right) \leq \sum_{i=1}^{n} P\left(A_{i}\right) $$

展开就是:

$$ \mathbb{P}\left(A_{1} \bigcup A_{2} \bigcup \cdots\right) \leq \mathbb{P}\left(A_{1}\right)+\mathbb{P}\left(A_{2}\right)+\cdots $$