probability density function

A probability density function is a probability distribution which maps every interval of values of a continuous random variable to its associated probability.

Formally,
a continuous random variable $X:\mathrm{\Omega }\to \mathbb{R}$ has probability density function $f(x)$ such that for any two $a\le b$ in $\mathbb{R}$,

where $f(x)$ is non-negative and ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=1$.

#WIP

The probability mass function of a discrete random variable can be defined as a probability density function against the counting measure.