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Linear Combination of Independent Exponential Random Variables

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Abstract

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.

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Acknowledgements

The work of the second author was done when he was with the Department of Electrical Engineering, Stanford University, USA.

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Authors and Affiliations

  1. Asian Cities Research Centre Ltd., Win Plaza, San Po Kong, Hong Kong

    Kim-Hung Li

  2. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, 94720, USA

    Cheuk Ting Li

Authors
  1. Kim-Hung Li
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Correspondence to Kim-Hung Li.

Appendix: Pseudo-Code for Computing Coefficient of H G,k(β i) for Given i:

Appendix: Pseudo-Code for Computing Coefficient of H G,k(β i) for Given i:

Let n be the size of β. Find m, the multiplicity of βiin β.

Find a which is the subvector of β with all m replications of β i removed.

Set J ← em,m.

If (m = n), return J as the vector of coefficients for (HG,1(βi),…,HG,m(βi)).

Set d ← βi.

For j = n − m downto 1, do { Set q ← 0.

For k = m downto 1, do { Set Jk ← (dJk − βiq)/(aj − βi). Set q ← Jk. }

Set d ← aj. }

Return (a1J/βi) as the vector of coefficients for (HG,1(βi),…,HG,m(βi)).

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Li, KH., Li, C.T. Linear Combination of Independent Exponential Random Variables. Methodol Comput Appl Probab 21, 253–277 (2019). https://doi.org/10.1007/s11009-018-9653-0

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  • DOI: https://doi.org/10.1007/s11009-018-9653-0

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