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Accueil du site > Séminaires > Probabilités Statistiques et réseaux de neurones > Joint distribution of the sum and maximum of iid exponential random variables

Vendredi 31 octobre 2008 à 11h00

Joint distribution of the sum and maximum of iid exponential random variables

Anna Panorska (University of Nevada, Reno, USA)

Résumé : We establish the joint distribution of the sum (X) and maximum (Y) of n iid exponential random variables. We derive exact formuli describing vector (X, Y) including its joint PDF, CDF, marginal and conditional distributions, moments, and the correlation structure. We also present new stochastic representations of (X, Y) and show its infinite divisibility and self-decomposability. These results lead to the exact distribution of the peak-to-average ratio nV, where V=X/Y, which surprisingly turns out independent of X. We briefly discuss parameter estimation for this new bivariate model. Additionally, we present a model with random sum and maximum (D, X, Y) where D has geometric distribution. Its bivariate marginals (D, X) and (D, Y) have already been applied in climate, hydrology and finance. The exact distribution of (X,Y) for finite n was of interest for many years. However, the only available results were of limiting nature. The distributions of (X, Y) and the peak-to-average ratio V are of keen interest in many applications including engineering, climatology, hydrology, finance, energy, and insurance. Our motivation for this work came from climate and hydrology research. We illustrate our models with an example from hydrology.

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