Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Probability density function ▷ Definition

Definition: Let $X$ be a continuous random variable with possible outcomes $\mathcal{X}$. Then, $f_X(x): \mathbb{R} \to \mathbb{R}$ is the probability density function (PDF) of $X$, if

\[\label{eq:pdf-def-s0} f_X(x) \geq 0\]

for all $x \in \mathbb{R}$,

\[\label{eq:pdf-def-s1} \mathrm{Pr}(X \in A) = \int_{A} f_X(x) \, \mathrm{d}x\]

for any $A \subset \mathcal{X}$ and

\[\label{eq:pdf-def-s2} \int_{\mathcal{X}} f_X(x) \, \mathrm{d}x = 1 \; .\]

Metadata: ID: D10 | shortcut: pdf | author: JoramSoch | date: 2020-02-13, 19:26.