Corollary 8.3.4. Alternate Form for the Exponential Distribution Probability Function.
Given a Poisson process and a constant \(\mu = \frac{1}{\lambda}\text{,}\) suppose \(X\) measures the variable interval length needed until you get a first success. Then \(X\) has an exponential distribution with probability function
\begin{equation*}
f(x) = \frac{1}{\mu} e^{-\frac{x}{\mu}}.
\end{equation*}