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:<math>F_X(x) = \operatorname{P}(X \le x)</math>
and sometimes also using a [[probability density function]], <math>f_X</math>. In [[measure theory|measure-theoretic]] terms, we use the random variable <math>X</math> to "push-forward" the measure <math>P</math> on <math>\Omega</math> to a measure <math>p_X</math> on <math>\mathbb{R}</math>. The measure <math>p_X</math> is called the "(probability) distribution of <math>X</math>" or the "law of <math>X</math>".
<ref name=":Billingsley">{{cite book|last1=Billingsley|first1=Patrick|title=Probability and Measure|date=1995|publisher=Wiley|edition=3rd|isbn=9781466575592|page=187}}</ref> The density <math>f_X = dp_X/d\mu</math>, the [[Radon–Nikodym derivative]] of <math>p_X</math> with respect to some reference measure <math>\mu</math> on <math>\mathbb{R}</math> (often, this reference measure is the [[Lebesgue measure]] in the case of continuous random variables, or the [[counting measure]] in the case of discrete random variables). The underlying probability space <math>\Omega</math> is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as [[correlation and dependence]] or [[Independence (probability theory)|independence]] based on a [[joint distribution]] of two or more random variables on the same probability space. In practice, one often disposes of the space <math>\Omega</math> altogether and just puts a measure on <math>\mathbb{R}</math> that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article on [[quantile function]]s for fuller development.
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