Definition:Probability Density Function/Naive Definition

Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X: \Omega \to \R$ be an absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $F_X: \R \to \closedint 0 1$ be the cumulative distribution function of $X$.

Let $\SS$ be the set of points at which $F_X$ is differentiable.

We define:

$\forall x \in \R: \map {f_X} x = \begin {cases}

\map {F_X'} x & : x \in \SS \\ 0 & : x \notin \SS \end {cases}$

where $\map {F_X'} x$ denotes the derivative of $F_X$ at $x$.