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 $\left({\Omega, \Sigma, \Pr}\right)$.

Then the (probability) density function or p.d.f. of $X$ is the function is the analog of the probability mass function for continuous $X$.

It is defined as:


 * $f_X: \R \to \left[{0 .. 1}\right]$:


 * $\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}$

Here $\Omega_X$ is defined as $\operatorname{Im} \left({X}\right)$, the image of $X$.