nLab law of a random variable

Idea

Given a probability space $(S,\mu)$ and a random variable $X \colon S \to \mathbb{R}$ with values in the real numbers, the law of $X$ is the probability measure on $\mathbb{R}$ which to a measurable subset assigns the probability that $X$ takes values in this subset.

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

Examples

Example

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

References

(…)