# Joint distribution of a random sample and an order statistic: A new approach with an application in reliability analysis

## TL;DR

This paper introduces a new approach to analyze the joint distribution of a sample and an order statistic, with applications in estimating inspection needs for reliability systems.

## Contribution

It presents a novel method for deriving the joint distribution of sample elements and order statistics, aiding reliability analysis and inspection planning.

## Key findings

- Derived formulas for joint distribution involving order statistics.
- Application to estimate inspection numbers in coherent systems.
- Enhanced reliability assessment techniques.

## Abstract

This paper considers the joint distribution of elements of a random sample and an order statistic of the same sample. \ The motivation for this work stems from the important problem in reliability analysis, to estimate the number of inspections we need in order to detect failed components in a coherent system. We consider an $(n-r+1)$-out-of-$n$ system, which is intact until at least $n-r+1$ of the components are alive, and it fails if the number of failed components exceeds $r$. The life time of the system is the $% r$th order statistic. Assuming that some of the components failed but the system \ is still functioning, \ using the results presented in this paper it is possible to find an expected value of the number of inspections we need to do for detecting certain number of failed components.