Order statistics leads to the specification of the statistics of a specific outcome based on a set of experimental outcomes that have been ordered, for example, from the lowest to the highest level. Ordering of experimental outcomes is of interest in many areas, and one prominent case is when the maximum outcome is required. Of interest to the development of material in this book is the ordering of the outcomes of a Poisson experiment to obtain an ordered set of times that underpin Poisson processes, shot noise random process, etc.
This chapter provides an introduction to ordered random variable theory including general results for the cumulative distribution function, probability density function, and joint probability density function of such random variables. The independent random variable case is assumed. As the difference between ordered outcomes is of importance to Poisson processes, the probability density function and joint probability density function of the difference random variables arising from ordering are considered. Specific results are established for both the finite and infinite intervals assuming that the underlying random variables have a uniform distribution. This assumption is consistent with the outcomes underpinning Poisson point and associated random processes. Useful references for order statistics include Arnold et al. (2008) and David and Nagaraja (2003).
11.1.1 Background Results
The following subsections ...