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

Result R3216 on D2325: Random euclidean real number characteristic function

Characteristic function of an almost constant random real number

Formulation 1

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	\begin{equation} \exists \, a \in \mathbb{R} : \mathbb{P}(X = a) = 1 \end{equation}

Then \begin{equation} \forall \, t \in \mathbb{R} : \mathfrak{F}_X(t) = e^{i t a} \end{equation}

Formulation 2

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	\begin{equation} \exists \, a \in \mathbb{R} : \mathbb{P}(X = a) = 1 \end{equation}

Then \begin{equation} \forall \, t \in \mathbb{R} : \mathbb{E} \left( e^{i t X} \right) = e^{i t a} \end{equation}

Proofs

Proof 0

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	\begin{equation} \exists \, a \in \mathbb{R} : \mathbb{P}(X = a) = 1 \end{equation}

If $t \in \mathbb{R}$, then \begin{equation} \mathfrak{F}_X (t) = \mathbb{E} \left( e^{i t X} \right) = e^{i t a} \mathbb{P}(X = a) = e^{i t a} \end{equation} $\square$