# 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 known as

Other words used to mean the same thing as **random variable** are:

**stochastic variable****chance variable****variate**.

The image $\Img X$ of $X$ is often denoted $\Omega_X$.

## Also see

- Results about
**continuous random variables**can be found**here**.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**continuous random variable**