Category:Continuous Random Variables

This category contains results about Continuous Random Variables.
Definitions specific to this category can be found in Definitions/Continuous Random Variables.

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

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$ such that the domain of $X$ is a continuum.

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

$\ds \lim_{\delta x \mathop \to 0} \map \Pr {X \in \openint x {x + \delta x} } = \map f x \delta x$

where $f$ is the frequency function on $X$.