ThmDex – An index of mathematical definitions, results, and conjectures.

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}

Proofs

Proof 0

Let $X, Y \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number.