# Characterization of Probability Density Function

## Theorem

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 $X$.

Let the probability density function of $X$ is the mapping $f_X: \R \to \closedint 0 1$ be defined as:

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

Suppose that the cumulative distribution function of $X$ defines a continuously differentiable real function $F_X: x \mapsto \map \Pr {X \le x}$.

Then the probability density function of $X$ is the mapping $f_X: \R \to \closedint 0 1$ satisfies:

$\dfrac \d {\d x} \map {F_X} x = \map {f_X} x$.

## Proof

 $\ds \map \Pr {x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2}$ $=$ $\ds \map {F_X} {x + \frac \epsilon 2} - \map {F_X} {x - \frac \epsilon 2} + \map \Pr {X = x -\frac \epsilon 2}$ $\ds$ $=$ $\ds \map {F_X} {x + \frac \epsilon 2} - \map {F_X} {x - \frac \epsilon 2}$ Probability of Continuous Random Variable at Single Point is Zero $\ds \leadsto \ \$ $\ds \map {f_X} x$ $=$ $\ds \lim_{\epsilon \mathop \to 0^+} \frac {\map {F_x} {x + \frac \epsilon 2} - \map {F_x} {x - \frac \epsilon 2} } \epsilon$ $\ds$ $=$ $\ds \dfrac \d {\d x} \map {F_X} x$ Continuous Derivative as Average of One-Sided Derivatives

$\blacksquare$