The hyperexponential (mixture exponential) distribution may be useful for fitting fat-tailed empirical distributions. \subsection{Hyperexponential densities} The hyperexponential distribution is a mixture of $K$ independent exponentials with a common location $\mu$ and different decay constants $\lambda_k$. The probability density function (PDF) is: \begin{equation} P(X=x) = \sum_k^{K} q_k \lambda_k e^{- \lambda_k (x - \mu)} \label{eqn:hyperexp_pdf} \end{equation} The cumulative distribution function (CDF) is: \begin{equation} P(X \leq x) = \sum_k^{K} q_k (1 - e^{- \lambda_k (x - \mu)}) \label{eqn:hyperexp_cdf} \end{equation} Variate $x$ ranges $\mu \leq x < \infty$. Mixture coefficients $q_k$ specify the prior probability of each component $k$; $0 \leq q_k \leq 1$ and $\sum_k q_k = 1$. The single location parameter $\mu$ is unconstrained, $-\infty < \mu < \infty$. (Exponential distributions are usually represented without an explicit location parameter, implicitly assuming $\mu = 0$.) The scale parameters $\lambda_k$ for each component are nonnegative, $\lambda_k > 0$.