ThmDex – An index of mathematical definitions, results, and conjectures.
Result R2405 on D2325: Random euclidean real number characteristic function
Subresult of R4391: Characteristic function uniquely identifies the distribution of a random euclidean real number
Characteristic function uniquely identifies the distribution of a random real number
Formulation 0
Let $X, Y \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number.
Then \begin{equation} \forall \, t \in \mathbb{R} : \mathbb{E}(e^{i t X}) = \mathbb{E}(e^{i t Y}) \quad \iff \quad X \overset{d}{=} Y \end{equation}
Formulation 1
Let $X, Y \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number.
Then \begin{equation} \mathfrak{F}_X = \mathfrak{F}_Y \quad \iff \quad X \overset{d}{=} Y \end{equation}
Also known as
Two random real numbers equal in distribution iff characteristic functions coincide, Characteristic function identifies distribution of random real numbers
Proofs
Proof 0
Let $X, Y \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number.