In probability theory, the Chung–Erdős inequality provides a lower bound on the probability that one out of many (possibly dependent) events occurs. The lower bound is expressed in terms of the probabilities for pairs of events.
Formally, let be events. Assume that for some . Then
![{\displaystyle \Pr[A_{i}]>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5980f0cd4c4a7e6cca9a665a65844f2a29379e)
![{\displaystyle \Pr[A_{1}\vee \cdots \vee A_{n}]\geq {\frac {\left(\sum _{i=1}^{n}\Pr[A_{i}]\right)^{2}}{\sum _{i=1}^{n}\sum _{j=1}^{n}\Pr[A_{i}\wedge A_{j}]}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2a27a07472d2a53ad395ddfc26566eb4e0f6a67)
The inequality was first derived by Kai Lai Chung and Paul Erdős (in,[1] equation (4)). It was stated in the form given above by Petrov (in,[2] equation (6.10)).
References edit
- ^ Chung, K. L.; Erdös, P. (1952-01-01). "On the application of the Borel–Cantelli lemma". Transactions of the American Mathematical Society. 72 (1): 179–186. doi:10.1090/S0002-9947-1952-0045327-5. ISSN 0002-9947.
- ^ Petrov, Valentin Vladimirovich (1995-01-01). Limit theorems of probability theory : sequences of independent random variables. Clarendon Press. OCLC 301554906.
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