ThmDex – An index of mathematical definitions, results, and conjectures.
Real calculus expression for distribution function of standard exponential random positive real number
Formulation 0
Let $T \in \text{Exp}(\theta)$ be a D4000: Standard exponential random positive real number.
Let $t \in \mathbb{R}$ be a D993: Real number.
Then \begin{equation} \mathbb{P}(T \leq t) = \left( 1 - e^{- t} \right) I_{[0, \infty)} (t) \end{equation}
Formulation 1
Let $T \in \text{Exp}(\theta)$ be a D4000: Standard exponential random positive real number.
Let $t \in \mathbb{R}$ be a D993: Real number.
Then \begin{equation} \mathbb{P}(T \leq t) = \left( 1 - e^{- t} \right) I_{t \in [0, \infty)} \end{equation}
Proofs