# Definition:Random Variable/Continuous/Absolutely Continuous

## Definition

### Definition 1

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $P_X$ be the probability distribution of $X$.

Let $\map \BB \R$ be the Borel $\sigma$-algebra of $\R$.

Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.

We say that $X$ is an absolutely continuous random variable if and only if:

$P_X$ is absolutely continuous with respect to $\lambda$.

### Definition 2

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $F_X$ be the cumulative distribution function of $X$.

We say that $X$ is an absolutely continuous random variable if and only if:

$F_X$ is absolutely continuous.

## Also known as

Often, particularly in elementary discussions of probability theory, the term continuous random variable is used to mean absolutely continuous random variable as defined here.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ we seek to avoid this ambiguity, and it should be made explicit whether a result applies only to absolutely continuous random variables, or general continuous random variables.

## Also see

• Results about absolutely continuous random variables can be found here.