Definition:Probability Density Function

Definition

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

Let $X: \Omega \to \R$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $\Omega_X = \Img X$, the image of $X$.

Then the probability density function of $X$ is the mapping $f_X: \R \to \R_{\ge 0}$ defined as:

$\forall x \in \R: \map {f_X} x = \begin {cases} \ds \lim_{\epsilon \mathop \to 0^+} \frac {\map \Pr {x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2} } \epsilon & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end {cases}$

Also known as

Probability density function is often conveniently abbreviated as p.d.f. or pdf.

Sometimes it is also referred to as the density function.

Also see

• Definition:Probability Mass Function

Sources

• 2001: Michael A. Bean: Probability: The Science of Uncertainty: $\S 4.1$