# Definition:Probability Density Function

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## Definition

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

Let $X$ be an absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $P_X$ be the probability distribution of $X$.

Let $\map \BB \R$ be the Borel $\sigma$-algebra of $\R$.

Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.

We define the **probability density function** $f_X$ by:

- $\ds f_X = \frac {\d P_X} {\d \lambda}$

where $\dfrac {\d P_X} {\d \lambda}$ denotes the Radon-Nikodym derivative of $P_X$ with respect to $\lambda$.

### Naïve Definition

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

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

Let $F_X: \R \to \closedint 0 1$ be the cumulative distribution function of $X$.

Let $\SS$ be the set of points at which $F_X$ is differentiable.

We define:

- $\forall x \in \R: \map {f_X} x = \begin {cases} \map {F_X'} x & : x \in \SS \\ 0 & : x \notin \SS \end {cases}$

where $\map {F_X'} x$ denotes the derivative of $F_X$ at $x$.

## 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**.

It is also known as a **frequency function**, which is also used for **probability mass function**

## Also see

- Results about
**probability density functions**can be found**here**.

## Sources

- 1991: David Williams:
*Probability with Martingales*... (previous) ... (next): $6.12$: The 'elementary formula' for expectation - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**frequency function** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**frequency function** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**probability density function**