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

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