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In
probability theory Probability theory or probability calculus 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 expre ...
, the general form of Bienaymé's identity, named for Irénée-Jules Bienaymé, states that :\operatorname\left( \sum_^n X_i \right)=\sum_^n \operatorname(X_i)+2\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
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...
or just
uncorrelated In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, ther ...
, integrable
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s, each with finite second moment. This simplification gives: :\operatorname\left(\sum_^n X_i\right) = \sum_^n \operatorname(X_k). The above expression is sometimes referred to as Bienaymé's formula. Bienaymé's identity may be used in proving certain variants of the
law of large numbers In probability theory, the law of large numbers is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law o ...
.


See also

*
Variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
* Propagation of error * Markov chain central limit theorem * Panjer recursion * Inverse-variance weighting * Donsker's theorem * Paired difference test


References

{{DEFAULTSORT:Bienayme's identity Algebra of random variables