# Category:Absolutely Continuous Random Variables

This category contains results about **absolutely continuous random variables**.

### 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.

## Pages in category "Absolutely Continuous Random Variables"

The following 3 pages are in this category, out of 3 total.