Characterization of Probability Density Function

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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

$\blacksquare$


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