Definition: Joint probability density function

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability density function ▷ Joint probability density function

Definition: Let $X \in \mathbb{R}^{n}$ be a continuous random vector. Then, a function $f_X(x): \mathbb{R}^n \to \mathbb{R}$ is the joint probability density function of $X$, if

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

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

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

for any $A \subset \mathbb{R}^n$ and

\[\label{eq:pdf-joint-def-s2} \int_{\mathbb{R}^n} f_X(x) \, \mathrm{d}x = 1 \; .\]

 

Sources:

• Wikipedia (2025): "Probability density function"; in: Wikipedia, the free encyclopedia, retrieved on 2025-02-06; URL: https://en.wikipedia.org/wiki/Probability_density_function#Densities_associated_with_multiple_variables.

Metadata: ID: D216 | shortcut: pdf-joint | author: JoramSoch | date: 2025-02-06, 16:36.