ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5376 on D1761: Real Riemann integral
Subresult of R5375: Riemann integral of decaying standard natural real exponential function over the positive real line
Riemann integral of an exponential random positive real number density function
Formulation 0
Let $f : [0, \infty) \to (0, \infty)$ be a D5482: Positive real function such that
(i) $\theta \in (0, \infty)$ is a D5407: Positive real number
(ii) \begin{equation} f(x) = \theta e^{- \theta x} \end{equation}
Then
(1) $f$ is an D6103: Improperly Riemann integrable real function on $[0, \infty)$
(2) \begin{equation} \int^{\infty}_0 f(t) \, d t = 1 \end{equation}
Proofs
Proof 0
Let $f : [0, \infty) \to (0, \infty)$ be a D5482: Positive real function such that
(i) $\theta \in (0, \infty)$ is a D5407: Positive real number
(ii) \begin{equation} f(x) = \theta e^{- \theta x} \end{equation}