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Definition D2325
Random euclidean real number characteristic function

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 $$\mathbb{R}^N \to \mathbb{C}, \quad \xi \mapsto \mathbb{E}_{\mathbb{P}}(e^{i t \cdot X})$$

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 $$\mathbb{R}^N \to \mathbb{C}, \quad \xi \mapsto \int_{\Omega} e^{i t \cdot X} \, d \mathbb{P}$$
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