Random Variable

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Definition (Random Variable). Let $(\Omega ,\mathcal{F},P)$ be a probability triple, $(\mathcal{X},\mathcal{S})$ be a measurable space. The map $X:\Omega \to \mathcal{X}$ is said to be a random variable, if $X$ is a measurable function. The probability that $X$ takes on a value in a measurable set $S\subseteq \mathcal{X}$ is written as

{\displaystyle \operatorname {P} (X\in S)=\operatorname {P} (\{\omega \in \Omega \mid X(\omega )\in S\})}\;. $$When $\mathcal{S}=\mathbb{R}$, a [[Random Variable|random variable]] $X:\Omega\to\mathbb{R}$ is said to be a continuous [[Random Variable|random variable]] or a real-valued random variable. ## References [Random variable - Wikipedia](https://en.wikipedia.org/wiki/Random_variable) [[D. Ostwald - The general linear model 20_21]]