Editing Random variable (section)

==Distribution functions==

If a random variable <math>X\colon \Omega \to \mathbb{R}</math> defined on the probability space <math>(\Omega, \mathcal{F}, \operatorname{P})</math> is given, we can ask questions like "How likely is it that the value of <math>X</math> is equal to 2?". This is the same as the probability of the event <math>\{ \omega : X(\omega) = 2 \}\,\! </math> which is often written as <math>P(X = 2)\,\!</math> or <math>p_X(2)</math> for short.

Recording all these probabilities of outputs of a random variable <math>X</math> yields the [[probability distribution]] of <math>X</math>. The probability distribution "forgets" about the particular probability space used to define <math>X</math> and only records the probabilities of various output values of <math>X</math>. Such a probability distribution, if <math>X</math> is real-valued, can always be captured by its [[cumulative distribution function]]

:<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.