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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. This inequality provides an upper bound on the probability of occurrence of at least one ...
In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
An important example, especially in the theory of probability, is the Borel algebra on the set of real numbers.It is the algebra on which the Borel measure is defined. . Given a real random variable defined on a probability space, its probability distribution is by definition also a measure on the Borel a
Union and intersection of Bloom filters with the same size and set of hash functions can be implemented with bitwise OR and AND operations, respectively. The union operation on Bloom filters is lossless in the sense that the resulting Bloom filter is the same as the Bloom filter created from scratch using the union of the two sets.
If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.. If (X, ≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongs to an upper set U and x ≤ y, then y must also belong to U) are closed under both intersections and unions.
However, the union of a fuzzy set and its complement may not result in the full universe U, and the intersection of them may not give the empty set ∅. Since the intersection and union are associative, it is natural to define the intersection and union of a finite family of fuzzy sets recursively. It is noteworthy that the generally accepted ...
Independence (probability theory) – When the occurrence of one event does not affect the likelihood of another; 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions; Monotone class theorem; Probability distribution – Mathematical function for the probability a given outcome occurs in an experiment
Then is a 𝜎-field over the set - to check closure under countable union, recall a -ring is closed under countable intersections. In fact A {\displaystyle {\mathcal {A}}} is the minimal 𝜎-field containing R {\displaystyle {\mathcal {R}}} since it must be contained in every 𝜎-field containing R . {\displaystyle {\mathcal {R}}.}