Probability Density Function

math probability statistics

Definition

Probability Density Function

A probability density function (PDF) of a continuous random variable $X$ is a non-negative function $f_X:\mathcal{X}\to {\mathbb{R}}_{\ge 0}$ such that the probability of $X$ taking a value in a measurable set $B$ is given by the Lebesgue integral:

$$P(X\in B)={\int }_B{f}_X(x)dx$$

For the distribution to be normalised, it must satisfy ${\int }_{\mathcal{X}}{f}_X(x)dx=1$. Unlike probability mass functions, $f_X(x)$ does not represent a probability and may exceed unity.