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
- Equivalence of Definitions of Absolutely Continuous Random Variable
- Absolutely Continuous Random Variable is Continuous
- Results about absolutely continuous random variables can be found here.