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

Characteristic function of exponential random positive real number

Formulation 1

Let $X \in \text{Exponential}(\theta)$ be an D214: Exponential random positive real number.
Let $t \in \mathbb{R}$ be a D993: Real number.

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

Formulation 2

Let $X \in \text{Exponential}(\theta)$ be an D214: Exponential random positive real number.
Let $t \in \mathbb{R}$ be a D993: Real number.

Then \begin{equation} \mathfrak{F}_X (t) = \frac{\theta}{\theta - i t} \end{equation}