### Order Statistics

Example 1: Continuous     Logistic distribution

Let random variable X have a Logistic distribution with pdf :

In:= Let denote a sample of size n drawn on X, and let denote the ordered sample, so that . The pdf of the r th order statistic, , is given by the mathStatica function:

In:= Out= The following diagram plots the pdf of the r th order statistic, as r increases from 1 to 10, given a sample of size n = 10: The joint pdf of and , for , is given by:

In:= Out:= A new feature in mathStatica 1.2 is that the OrderStat function now also supports discrete random variables. To illustrate, let random variable with pmf :

In:= Let denote a random sample of size n drawn on X, and let denote the order statistics.

Then, the pmf of the r th order statistic, , denoted , is given immediately by:

In:= Out:= with domain of support:

In:= Out:= The following diagram plots the pmf of the r th order statistic that we have just derived, when , and r = 1 and the sample size is n = 10:

In:=  