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

Characteristic function of standard Poisson random natural number

Formulation 0

Let $X \in \text{Poisson}(1)$ be a D5524: Standard Poisson random natural number.
Let $t \in \mathbb{R}$ be a D993: Real number.

Then \begin{equation} \mathbb{E}(e^{i t X}) = e^{(e^{i t} - 1)} \end{equation}

Formulation 1

Let $X \in \text{Poisson}(1)$ be a D5524: Standard Poisson random natural number.
Let $t \in \mathbb{R}$ be a D993: Real number.

Then \begin{equation} \mathfrak{F}_X(t) = \exp\left( e^{i t} - 1 \right) \end{equation}

Proofs

Proof 0

Let $X \in \text{Poisson}(1)$ be a D5524: Standard Poisson random natural number.
Let $t \in \mathbb{R}$ be a D993: Real number.

This result is a particular case of R3904: Characteristic function of a Poisson random natural number. $\square$