# Lesson 46: Order Statistics

### Introduction

We typically don't pay particular attention to the order of a set of random variables *X*_{1}, *X*_{2}, .., *X _{n}*. But, what if we did? Suppose, for example, we needed to know the probability that the third largest value was less than 72. Or, suppose we needed to know the 80th percentile of a random sample of heart rates. In either case, we'd need to know something about how the

*order*of the data behaved. That is, we'd need to know something about the probability density function of the

**order statistics**

*Y*

_{1},

*Y*

_{2}, ...,

*Y*. That's what we'll groove on in this lesson.

_{n}### Objectives

- To learn the formal definition of order statistics.
- To derive the distribution function of the
*r*order statistic.^{th} - To derive the probability density function of the
*r*order statistic.^{th} - To derive a method for finding the (100
*p*)percentile of the sample.^{th}