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| In [[probability]] and [[statistics]], a '''realization''', or '''observed value''', of a [[random variable]] is the value that is actually observed (what actually happened). The random variable itself should be thought of as the process how the observation comes about. Statistical quantities computed from realizations without deploying a statistical model are often called "[[empirical]]", as in [[empirical distribution function]] or [[empirical probability]].
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| Conventionally, upper case letters denote random variables; the corresponding lower case letters denote their realizations.<ref name="Wilks-1962-MS"> Samuel S. Wilks. ''Mathematical statistics''. A Wiley Publication in Mathematical Statistics. John Wiley & Sons Inc., New York, 1962.</ref> Confusion results when this important convention is not strictly observed.
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| In more formal [[probability theory]], a random variable is a [[Function (mathematics)|function]] ''X'' defined from a [[sample space]] Ω to a '''measurable space''' called the '''state space'''.<ref name="Varadhan-2001-PT"> S. R. S. Varadhan. ''Probability theory'', volume 7 of ''Courant Lecture Notes in Mathematics''. New York University Courant Institute of Mathematical Sciences, New York, 2001.</ref> If an element in Ω is sent to an element in state space by ''X'', then that element in state space is a realization. (In fact, a [[random variable]] cannot be an arbitrary function and it needs to satisfy another condition: it needs to be [[measurable]].) Elements of the sample space can be thought of as all the different possibilities that '''could''' happen; while a realization (an element of the state space) can be thought of as the value ''X'' attains when one of the possibilities '''did''' happen. [[Probability]] is a [[Map (mathematics)|mapping]] that assigns numbers between zero and one to certain [[subset]]s of the sample space. Subsets of the sample space that contain only one element are called [[elementary event]]s. The value of the random variable (that is, the function) ''X'' at a point ω ∈ Ω,
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| :<math> x = X(\omega)</math>
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| is called a '''realization''' of ''X''.
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| ==References==
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| <references/>
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| [[Category:Statistical terminology]]
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Latest revision as of 17:19, 24 May 2014
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