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

Result R5376 on D1761: Real Riemann integral

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}