# Expected value

## Average value of a random variable / From Wikipedia, the free encyclopedia

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In probability theory, the **expected value** (also called **expectation**, **expectancy**, **expectation operator**, **mathematical expectation**, **mean**, **average**, or **first moment**) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would "expect" to get in reality.

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The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration.

The expected value of a random variable X is often denoted by E(*X*), E[*X*], or E*X*, with E also often stylized as E or $\mathbb {E} .$[1][2][3]