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In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. [1] Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not mutually independent.
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies measure properties such as countable additivity. [1] The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume ) is that a probability measure must ...
The probability measure function must satisfy two simple requirements: ... is a countable collection of pairwise disjoint sets, then (=) = = (), the ...
Pairwise generally means "occurring in pairs" or "two at a time." Pairwise may also refer to: Pairwise disjoint; Pairwise independence of random variables; Pairwise comparison, the process of comparing two entities to determine which is preferred; All-pairs testing, also known as pairwise testing, a software testing method.
Part of a series on statistics: Probability theory; Probability. Axioms; Determinism. System; Indeterminism; Randomness; Probability space; Sample space; Event ...
In probability theory, the joint probability distribution is the probability distribution of all possible pairs of outputs of two random variables that are defined on the same probability space. The joint distribution can just as well be considered for any given number of random variables.
We want to bound the probability that any is a subset of . We will bound it using the expectation of the number of A ∈ S {\displaystyle A\in S} such that A ⊆ Γ p {\displaystyle A\subseteq \Gamma _{p}} , which we call λ {\displaystyle \lambda } , and a term from the pairwise probability of being in Γ p {\displaystyle \Gamma _{p}} , which ...