Bienaymé's Identity
In probability theory, the general form of Bienaymé's identity states that :\operatorname\left( \sum_^n X_i \right)=\sum_^n \operatorname(X_i)+\sum_^n \operatorname(X_i,X_j)=\sum_^n\operatorname(X_i,X_j). This can be simplified if X_1, \ldots, X_n are pairwise Independence (probability theory), independent or just Uncorrelatedness (probability theory), uncorrelated, integrable random variables, each with finite second Moment (mathematics), moment. This simplification gives: :\operatorname\left(\sum_^n X_i\right) = \sum_^n \operatorname(X_k). Bienaymé's identity may be used in proving certain variants of the law of large numbers. See also *Propagation of error *Markov chain central limit theorem References {{DEFAULTSORT:Bienayme's identity Algebra of random variables ...
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Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ...
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