ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Operation
N-operation
Binary operation
Enclosed binary operation
Groupoid
Ringoid
Semiring
Ring
Left ring action
Module
Vector space
Vector space seminorm
Vector space norm
Normed vector space
Bounded set
Bounded map
Constant-bounded map
Constant-bounded function
Finite measure
Finite unsigned euclidean real Borel measure Fourier transform
Probability measure fourier transform
Random euclidean complex number characteristic function
Definition D2325
Random euclidean real number characteristic function
Formulation 4
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X : \Omega \to \mathbb{R}^N$ is a D4383: Random euclidean real number on $P$
The characteristic function of $X$ is the D992: Function \begin{equation} \mathbb{R}^N \to \mathbb{C}, \quad \xi \mapsto \mathbb{E}_{\mathbb{P}}(e^{i t \cdot X}) \end{equation}
Formulation 5
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X : \Omega \to \mathbb{R}^N$ is a D4383: Random euclidean real number on $P$
The characteristic function of $X$ is the D992: Function \begin{equation} \mathbb{R}^N \to \mathbb{C}, \quad \xi \mapsto \int_{\Omega} e^{i t \cdot X} \, d \mathbb{P} \end{equation}
Results
Characteristic function of almost surely constant random euclidean real number
Characteristic function for translated random real number
Characteristic function of Bernoulli random boolean number
Characteristic function of a Poisson random natural number
Characteristic function of a binomial random natural number
Characteristic function of a scaled random real number
Characteristic function of gaussian random real number
Characteristic function of geometric random positive integer
Characteristic function of indicator random boolean number
Characteristic function of rademacher random integer
Characteristic function of standard Poisson random natural number
Characteristic function of standard gaussian random real number
Characteristic function uniquely identifies the distribution of a random real number