March, 1949 A Modified Extreme Value Problem
Benjamin Epstein
Ann. Math. Statist. 20(1): 99-103 (March, 1949). DOI: 10.1214/aoms/1177730095

## Abstract

Consider the following problem. Particles are distributed over unit areas in such a way that the number of particles to be found in such areas is a random variable following the law of Poisson, with $\nu$ equal to the expected number of particles per unit area. Furthermore, the particles themselves are assumed to vary in magnitude according to a size distribution specified (independently of the particular unit area chosen) by a d.f. $F(x)$ defined over some interval $a \leq x \leq b$, with $F(a) = 0$ and $F(b) = 1$. The problem is to find the distribution of the smallest, largest, or more generally the $n$th smallest or $n$th largest particle in randomly chosen unit areas. The problem as stated is not completely specified. To specify the distribution of smallest or largest particles in a unit area one must give a rule for dealing with those areas which contain no particles at all. More generally, in the case of the distribution of the $n$th smallest or $n$th largest particle, one must give a rule for dealing with those areas which contain $(n - 1)$ or fewer particles. There are at least two possible alternatives. One alternative is to omit none of the areas from consideration by setting up the following rule: if no particles are found in a given unit area then this area will be considered as one for which the smallest size particle is $x = b$ and for which the largest size particle is $x = a$. More generally, if $(n - 1)$ or fewer particles are found in a given unit area then this area will be considered as one for which the $n$th smallest size particle is $x = b$ and for which the $n$th largest size particle is $x = a$. A second alternative is to restrict attention to those areas which contain at least one particle (in the case of the distribution of smallest or largest values) or at least $n$ particles (in the case of the distribution of the $n$th smallest or $n$th largest particle). In other words, this means finding the relevant conditional distribution. From the point of view of the application of the theory of extreme values to fracture problems, there are some situations where the first model and other situations where the second model is the more appropriate in describing the phenomenon under investigation. In this paper section 2 will be devoted to a derivation of the distributions associated with the first alternative; in section 3 the conditional distributions will be described briefly.

## Citation

Benjamin Epstein. "A Modified Extreme Value Problem." Ann. Math. Statist. 20 (1) 99 - 103, March, 1949. https://doi.org/10.1214/aoms/1177730095

## Information

Published: March, 1949
First available in Project Euclid: 28 April 2007

zbMATH: 0041.44902
MathSciNet: MR29114
Digital Object Identifier: 10.1214/aoms/1177730095