Mean Absolute Deviation for Hyperexponential and Hypoexponential Distributions

Hyperexponential and Hypoexponential distributions are derived from mixtures and convolutions of independent exponential random variables, respectively, and have a wide range of applications in telecommunications, quantitative finance, and reliability analysis. In addition, Bernstein's theorem states that all completely monotonic probability distribution functions (PDFs) can be expressed as mixtures of exponential distributions. In this paper, we not only explore these distributions but also pioneer the derivation of Mean Absolute Deviation (MAD) for them. We establish new Chebyshev-type bounds and Peek bounds that further enhance our understanding and exploitation of these distributions. Our contribution lies in providing explicit formulas for MAD calculation specific to Hyperexponential and Hypoexponential distributions and using the MAD in real-life applications.