# Definition:Random Variable/Continuous

## Definition

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

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

We say that $X$ is a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$ if and only if:

the cumulative distribution function of $X$ is continuous.

## Absolutely Continuous Random Variable

### 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.

## Singular Random Variable

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

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

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

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

We say that $X$ is singular if and only if:

there exists a $\lambda$-null set $B$ such that $\map \Pr {X \in B} = 1$.

## Also see

• Results about continuous random variables can be found here.