2.1. Random Variables

Random Variable

A random variable is a real-valued function $X$: $\text{Upomega}\to \mathbb{R}$ that assigns to each sample point (outcome) $\text{upomega}\in \text{Upomega}$ a real number, $X(\text{upomega})\in \mathbb{R}$.

• Discrete random variables feature a countable range \mathcal{X} = \set{X(\upomega): \upomega \in \Upomega}, such that for every $x\in \mathcal{X}$, the set
  \set{\upomega \in \Upomega: X(\upomega) = x} \in \mathcal{F}
  is an event in the sample space containing all the outcomes where the random variable $X$ equals the specific value $x\in \mathcal{X}$.
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• Continuous random variables can take on any value within a given interval of the continuous sample space, such that they feature an uncountable range $\mathcal{X}=[a,b]$ for some bounds $a,b\in \mathbb{R}$.