nLab law of a random variable

Idea

Given a probability space $(S,\mathcal{F},\mu)$ and a random variable or random element $f \colon S \to X$, the law or distribution of $f$ (or sometimes, of $X$) is the probability measure on $X$ which to a measurable subset $A\subseteq X$ assigns the probability that $f$ takes values in $A$.

In particular, the law of $f$ is the pushforward of $\mu$ along $f$.

Examples

Example

The law of a deterministic random variable is a Dirac measure.

See also

• random variable
• probability space
• probability monad
• joint and marginal probability

category: probability