Definition:Probability Density Function

Definition
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

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

Let $\Omega_X = \operatorname{Im} \left({X}\right)$, the image of $X$.

Then the probability density function of $X$ is the mapping :$f_X: \R \to \left[{0 \,.\,.\, 1}\right]$ defined as:


 * $\forall x \in \R: f_X \left({x}\right) = \begin{cases}

\displaystyle \lim_{\epsilon \to 0^+} \frac{\Pr \left({x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2}\right)} \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