Editing Expected value (section)

{{Short description|Average value of a random variable}}
{{About|the term used in probability theory and statistics}}
{{Redirect|E(X)|the <math>e^x</math> function|Exponential function}}
{{Probability fundamentals}}

In [[probability theory]], the '''expected value''' (also called '''expectation''', '''expectancy''', '''expectation operator''', '''mathematical expectation''', '''mean''', '''expectation value''', or '''first [[Moment (mathematics)|moment]]''') is a generalization of the [[weighted average]]. Informally, the expected value is the [[arithmetic mean|mean]] of the possible values a [[random variable]] can take, weighted by the probability of those outcomes. 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.

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 [[Integral|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 {{mvar|X}} is often denoted by {{math|E(''X'')}}, {{math|E[''X'']}}, or {{math|E''X''}}, with {{math|E}} also often stylized as <math>\mathbb{E}</math> or {{math|''E''}}.<ref>{{Cite web|title=Expectation {{!}} Mean {{!}} Average|url=https://www.probabilitycourse.com/chapter3/3_2_2_expectation.php|access-date=2020-09-11|website=www.probabilitycourse.com}}</ref><ref>{{Cite web|last=Hansen|first=Bruce|title=PROBABILITY AND STATISTICS FOR ECONOMISTS|url=https://ssc.wisc.edu/~bhansen/probability/Probability.pdf|access-date=2021-07-20|archive-date=2022-01-19|archive-url=https://web.archive.org/web/20220119041716/https://ssc.wisc.edu/~bhansen/probability/Probability.pdf|url-status=dead}}</ref><ref>{{cite book |last1=Wasserman |first1=Larry |title=All of Statistics: a concise course in statistical inference |date=December 2010 |publisher=Springer texts in statistics |isbn=9781441923226 |page=47}}</ref>

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