Law of the unconscious statistician
Theorem in probability and statistics / From Wikipedia, the free encyclopedia
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In probability theory and statistics, the law of the unconscious statistician, or LOTUS, is a theorem which expresses the expected value of a function g(X) of a random variable X in terms of g and the probability distribution of X.
The form of the law depends on the type of random variable X in question. If the distribution of X is discrete and one knows its probability mass function pX, then the expected value of g(X) is
where the sum is over all possible values x of X. If instead the distribution of X is continuous with probability density function fX, then the expected value of g(X) is
Both of these special cases can be expressed in terms of the cumulative probability distribution function FX of X, with the expected value of g(X) now given by the Lebesgue–Stieltjes integral
In even greater generality, X could be a random element in any measurable space, in which case the law is given in terms of measure theory and the Lebesgue integral. In this setting, there is no need to restrict the context to probability measures, and the law becomes a general theorem of mathematical analysis on Lebesgue integration relative to a pushforward measure.