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:

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

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

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

$\blacksquare$


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