cf.md

layout	mathjax	author	affiliation	e_mail	date	title	chapter	section	topic	definition	sources	def_id	shortcut	username
definition	true	Joram Soch	BCCN Berlin	joram.soch@bccn-berlin.de	2021-09-22 02:20:00 -0700	Characteristic function	General Theorems	Probability theory	Other probability functions	Characteristic function	authors year title in pages url Wikipedia 2021 Characteristic function (probability theory) Wikipedia, the free encyclopedia retrieved on 2021-09-22 https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)#Definition authors year title in pages url Taboga, Marco 2017 Joint characteristic function Lectures on probability and mathematical statistics retrieved on 2021-10-07 https://www.statlect.com/fundamentals-of-probability/joint-characteristic-function	D159	cf	JoramSoch

Definition:

1. The characteristic function of a random variable $X \in \mathbb{R}$ 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 $X \in \mathbb{R}^n$ 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 $X \in \mathbb{R}^{n \times p}$ is

$$ \label{eq:cf-mat} \varphi_X(t) = \mathrm{E} \left[ e^{i , \mathrm{tr} \left( t^\mathrm{T}X \right)} \right], \quad t \in \mathbb{R}^{n \times p} ; . $$