Joseph Hoover
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In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events.
Learn how to use the union bound and its extension, the Bonferroni inequalities, to bound the probability of union of events. See examples of applications in random graphs and expected value of events.
Learn how to use the union bound to bound the probability of multiple events occurring, and how to apply Jensen's inequality and Hoe ding's inequality to convex functions. See examples, proofs, and slides from CSE312 course at UW.
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Boole's inequality (or the union bound ) states that for any at most countable collection of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the events in the collection.
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The beautiful thing about the union bound is that it is extremely general—it applies whether the events are independent or not. In cryptography, we use the union bound all of the time. One common application is to define “bad” eventsB 1,...,B nand then to use the union bound to bound the probability that any of them occurs. 2 Linearity of ...
Any bound of this form is called a tail bound or concentration inequality. Today we will see three methods that give progressively stronger bounds, but under progressively stronger assumptions. They are Markov’s inequality, Chebyshev’s inequality, and the Cher-no bound. 2 Markov’s Inequality
The second inequality follows from symmetry and the last one using the union bound: IP(|Z| >t) = IP({Z>t}∪{Z< −t}) ≤ IP(Z>t)+IP(Z< −t) = 2IP(Z>t). The fact that a Gaussian random variable Z has tails that decay to zero exponentially fast can also be seen in the moment generating function (MGF)
