Linear Combination of Independent Exponential Random Variables
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In this paper we prove a recursive identity for the cumulative distribution function of a linear combination of independent exponential random variables. The result is then extended to probability density function, expected value of functions of a linear combination of independent exponential random variables, and other functions. Our goal is on the exact and approximate calculation of the above mentioned functions and expected values. We study this computational problem from different views, namely as a Hermite interpolation problem, and as a matrix function evaluation problem. Examples are presented to illustrate the applicability and performance of the methods.
KeywordsAffine combination Erlang distribution Hypoexponential distribution Hermite interpolating polynomial Matrix function Recurrence relation
Mathematics Subject Classification (2010)65Q30 65C50
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The work of the second author was done when he was with the Department of Electrical Engineering, Stanford University, USA.
- Cox DR (1962) Renewal Theory. Methuen and CoGoogle Scholar
- Goulet V, Dutang C, Maechler M, Firth D, Shapira M, Stadelmann M, expm-developers@lists.R-forge.R-project.org (2015) expm: Matrix Exponential. R package version 0.999-0. https://CRAN.R-project.org/package=expm
- Higham NJ (2008) Functions of matrices: theory and computation. SIAMGoogle Scholar
- Kadri T, Smaili K, Kadry S (2015) Markov modeling for reliability analysis using hypoexponential distribution. In: Kadry S, El Hami A (eds) Numerical methods for reliability and safety assessment: multiscale and multiphysics systems. Springer, Switzerland, pp 599–620Google Scholar
- Kotz S, Kozubowski TJ, Podgórski K (2001) The laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Birkhäuser, BostonGoogle Scholar
- Maechler M (2015) Rmpfr: R MPFR - multiple precision Floating-Point reliable. R package version 0.6–0. https://CRAN.R-project.org/package=Rmpfr
- R Core Team (2016) R: A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria. https://www.R-project.org/
- Van Loan CF (1975) A study of the matrix exponential. Numerical Analysis Report No. 10 Department of Mathematics. University of Manchester, EnglandGoogle Scholar
- Wen YZ, Yin CC (2014) A generalized Erlang(n) risk model with a hybrid dividend strategy (in Chinese). Sci Sin Math 44:1111–1122Google Scholar