Definition: Characteristic function

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Other probability functions ▷ Characteristic function

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

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

 

Sources:

• Wikipedia (2021): "Characteristic function (probability theory)"; in: Wikipedia, the free encyclopedia, retrieved on 2021-09-22; URL: https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)#Definition.
• Taboga, Marco (2017): "Joint characteristic function"; in: Lectures on probability and mathematical statistics, retrieved on 2021-10-07; URL: https://www.statlect.com/fundamentals-of-probability/joint-characteristic-function.

Metadata: ID: D159 | shortcut: cf | author: JoramSoch | date: 2021-09-22, 09:20.