A Qualitative Characterization of the Exponential Distribution

Canadian Journal of Statistics, 1993

A concept of the lack-of-memory property at a given time point c > 0 is introduced. It is equivalent to the concept of the almost-lack-of-memory (ALM) property of the random variables. A representation theorem is given for the cumulative distribution function of such random variables as well as for corresponding decompositions in terms of independent random variables. It is shown that a periodic failure rate for a random variable is equivalent to the ALM property. In addition some properties of the service time of an unreliable server are observed. RESUME On introduit la propriktk d'ktre "sans memoire" en un instant donnt c > 0. Ceci est kquivaleut au concept de variable alkatoire "presque sans mkmoire". On dkmontre un thkoreme de reprksentation pour la fonction de ripartition de telles variables aleatoires ainsi que pour les dkcompositions correspondantes en variables aleatoires indkpendantes. I1 est demonre qu'un taux de panne pkriodeque est une condition neccessaire et suffisante pour qu'une variable alkatoire soit "presque sans mCmoire".Finalement, on Ctudie quelques propriktes des temps de service d'un serveur non fiable.

A class of probability distributions is characterized via equalities in law between two order statistics shifted by independent exponential variables. An explicit formula for the quintile function of the identified family of distributions is obtained. The results extend some known characterizations of exponential and logistic distributions.

Journal of Statistics Applications & Probability, 2013

Distributional relations of the form Y d = X + T where X, Y , and T are record values or order statistics and the random translator T is independent from X are considered. Characterizations of the exponential distribution when the ordered random variables are non-neighboring are proved. Corollaries for Pareto and power function distributions are also derived.

Annals of the Institute of Statistical Mathematics, 2003

Two characterizations of the exponential distribution based on equalities among order statistics in a random sample of size three are proved. This proves two conjectures stated recently in Arnold and Villaseñor [4].

Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1971

Statistics & Probability Letters, 2008

Let F be a natural exponential family (NEF) generated by a measure µ and X = (X 1 , . . . , X n ) a random sample with a common distribution belonging to F. Consider the set of order statistics X (1) ≤ X (2) ≤ · · · ≤ X (n) and let G r,n denote the family of distributions induced by the r-th order statistic X (r) , r = 1, . . . , n. The main problem of the paper, namely, the closedness of NEF's under the formation of order statistics, can be posed as follows:

Journal of Statistical Theory and Applications, 2014

Some new characterizations of probability distributions based on properties of record values shifted by independent exponential variables are obtained. Explicit expressions for the inverse functions of the corresponding variables are presented. Among others some characterizations of exponential distributions and limiting distributions of maximal and minimal extreme values are given.

Automated Deduction–CADE-21, 2007

In order to overcome the limitations of state-of-the-art simulation based probabilistic analysis, we propose to perform probabilistic analysis within the environment of a higher-order-logic theorem prover. The foremost requirement for conducting such analysis is the formalization of probability distributions. In this report, we present a methodology for the formalization of continuous probability distributions for which the inverse of the cumulative distribution function can be expressed in a closed mathematical form. Our methodology is primarily based on the formalization of the Standard Uniform random variable, cumulative distribution function properties and the Inverse Transform method. The report presents all this formalization using the HOL theorem prover. In order to illustrate the practical effectiveness of our methodology, the formalization of a few continuous probability distributions has also been included.

probstat.org.in

Paper received on 13 June 2009; revised, 23 September 2009; accepted, 18 October 2009. Abstract. A family of continuous probability distribution has been characterized through the difference of two conditional expectations, conditioned on a non-adjacent order statistics. Further, ...

2022

The sum of independent, but not necessary identically distributed, exponential random variables follows hypoexponential distribution. We study a situation when the rate parameters of the exponential variables are not all different from each other. We obtain a representation for the Laplace transform of the hypoexponential distribution in the case of two repeated parameter values. Applying this decomposition, we prove a characterization of the exponential distribution.