The Distributions of the Peak to Average and Peak to Sum Ratios Under Exponentiality

  • Tomasz J. Kozubowski
  • Anna K. Panorska
  • Fares Qeadan


Let E 1, E 2, , E N be independent and identically distributed exponential random variables, and let \(Y = \vee _{i = 1}^N\) and S = ∑ i = 1 N E i be their maximum and sum, respectively. We review distributional properties of the peak to sum and peak to average ratios, R = YS and \(\tilde{R} = Y/(S/N)\) respectively, with deterministic N, and provide an extension to the case where N is itself a random variable, independent of the {E j }. Our results include explicit formulas for the relevant density and distribution functions, which apply to any distribution of N, as well as a particular example with geometrically distributed N. An example from climatology shows modeling potential of these models.


Probability Density Function Flood Risk Management Exponential Random Variable Deterministic Number Potential Climate Change Impact 
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This work was supported in part by NSF grants ATM-0236898 and ATM-0503722.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Tomasz J. Kozubowski
    • 1
  • Anna K. Panorska
  • Fares Qeadan
  1. 1.Department of Mathematics and Statistics, MS 084University of NevadaRenoUSA

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