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

Result R3844 on D2325: Random euclidean real number characteristic function

Characteristic function of a scaled random real number

Formulation 0

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

Then \begin{equation} \mathfrak{F}_{a X}(t) = \mathfrak{F}_X (a t) \end{equation}

Proofs

Proof 0

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

We have \begin{equation} \mathfrak{F}_{a X} (t) = \mathbb{E} (e^{i t (a X)}) = \mathbb{E} (e^{i (t a) X}) = \mathfrak{F}_X (t a) \end{equation} $\square$