Random Variable

A function from sample space to real line.

Discrete

A discrete Random Variable is characterized by its probability mass function.

A pmf, $f_X$ is such that:

$f_X:\Omega \to {\mathbb{R}}_{\ge 0}$ ${\sum }_{x\in \Omega }{f}_X(x)=1$

Continuous

A continuous RV is characterized by its probability distribution function. A pdf $f_Y$ is such that:

$f_Y:\Omega =\mathbb{R}\to {\mathbb{R}}_{\ge 0}$

${\int }_{\Omega }{f}_Y(y)dy=1$

The sample space for cts rv’s is often the reals.