# Definition:Probability Density Function

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## Contents

## 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 $f_X: \R \to \closedint 0 1$ defined as:

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

## 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

## Sources

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