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.