added 1 definition

Author: JoramSoch

D/cf.md

@@ -0,0 +1,48 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-09-22 09:20:00
+
+title: "Characteristic function"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Probability functions"
+definition: "Characteristic function"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2021
+    title: "Characteristic function (probability theory)"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2021-09-22"
+    url: "https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)#Definition"
+
+def_id: "D159"
+shortcut: "cf"
+username: "JoramSoch"
+---
+
+
+**Definition:**
+
+1) The characteristic function of a [random variable](/D/rvar) $X \in \mathbb{R}$ is
+
+$$ \label{eq:cf-var}
+\varphi_X(t) = \mathrm{E} \left[ e^{itX} \right], \quad t \in \mathbb{R} \; .
+$$
+
+2) The characteristic function of a [random vector](/D/rvec) $X \in \mathbb{R}^n$ is
+
+$$ \label{eq:cf-vec}
+\varphi_X(t) = \mathrm{E} \left[ e^{i t^\mathrm{T}X} \right], \quad t \in \mathbb{R}^n \; .
+$$
+
+3) The characteristic function of a [random matrix](/D/rmat) $X \in \mathbb{R}^{n \times p}$ is
+
+$$ \label{eq:cf-mat}
+\varphi_X(t) = \mathrm{E} \left[ e^{i \, \mathrm{tr} \left( t^\mathrm{T}X \right)} \right], \quad t \in \mathbb{R}^{n \times p} \; .
+$$