Definition:Probability Density Function
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Definition
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 $X$.
Then the probability density function of $X$ is the mapping $f_X: \R \to \R_{\ge 0}$ defined as:
- $\forall x \in \R: \map {f_X} x = \begin {cases} \ds \lim_{\epsilon \mathop \to 0^+} \frac {\map \Pr {x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2} } \epsilon & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end {cases}$
Also known as
Probability density function is often conveniently abbreviated as p.d.f. or pdf.
Sometimes it is also referred to as the density function.
Also see
Sources
- 2001: Michael A. Bean: Probability: The Science of Uncertainty: $\S 4.1$