Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Random variables ▷ Univariate vs. multivariate

Definition: Let $X$ be a random variable with possible outcomes $\mathcal{X}$. Then,

• $X$ is called a two-valued random variable or random event, if $\mathcal{X}$ has exactly two elements, e.g. $\mathcal{X} = \left\lbrace E, \overline{E} \right\rbrace$ or $\mathcal{X} = \left\lbrace \mathrm{true}, \mathrm{false} \right\rbrace$ or $\mathcal{X} = \left\lbrace 1, 0 \right\rbrace$;

• $X$ is called a univariate random variable or random scalar, if $\mathcal{X}$ is one-dimensional, i.e. (a subset of) the real numbers $\mathbb{R}$;

• $X$ is called a multivariate random variable or random vector, if $\mathcal{X}$ is multi-dimensional, e.g. (a subset of) the $n$-dimensional Euclidean space $\mathbb{R}^n$;

• $X$ is called a matrix-valued random variable or random matrix, if $\mathcal{X}$ is (a subset of) the set of $n \times p$ real matrices $\mathbb{R}^{n \times p}$.

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Metadata: ID: D106 | shortcut: rvar-uni | author: JoramSoch | date: 2020-11-06, 03:47.