Random variable
Definition

A random variable X {\displaystyle X} is a measurable function X: Ω → E {\displaystyle X\colon \Omega \to E} from a sample space Ω {\displaystyle \Omega } as a set of possible outcomes to a measurable space E {\displaystyle E}. The technical axiomatic definition requires the sample space Ω {\displaystyle \Omega } to be a sample space of a probability triple (Ω, F, P) {\displaystyle (\Omega,{\mathcal {F}},\operatorname {P})} (see the measure-theoretic definition). A random variable is often denoted by capital Roman letters such as X, Y, Z, T {\displaystyle X,Y,Z,T}.

The probability that X {\displaystyle X} takes on a value in a measurable set S ⊆ E {\displaystyle S\subseteq E} is written as
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