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

Result R4639 on D2325: Random euclidean real number characteristic function

Characteristic function for translated random real number

Formulation 0

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number.
Let $\mu, t \in \mathbb{R}$ each be a D993: Real number.

Then \begin{equation} \mathfrak{F}_{X + \mu} (t) = e^{i t \mu} \mathfrak{F}_X (t) \end{equation}

Proofs

Proof 0

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number.
Let $\mu, t \in \mathbb{R}$ each be a D993: Real number.

By definition, we have $\mathfrak{F}_X (t) = \mathbb{E} (e^{i t X})$. Therefore \begin{equation} \mathfrak{F}_{X + \mu} (t) = \mathbb{E} (e^{i t (X + \mu)}) = \mathbb{E} (e^{i t X} e^{i t \mu}) = e^{i t \mu} \mathbb{E} (e^{i t X}) \end{equation} $\square$