random variable

For a probability space $(\Omega ,\mathcal{F},P)$, a function $X:(\Omega ,\mathcal{F})\to (\mathbb{R},\mathcal{B})$ is called a random variable if it is a measureable function. Note here $\mathcal{B}$ is the 20231218115557-borel_sigma_algebra on the real numbers. This is important, because we want random variables to be real-valued. Also, the Borel $\sigma$ -algebra has some very nice properties to use.

A set of random variables $\{X_1,\dots ,X_n\}$ is called a random vector.