1. Continuous Parent Distribution

Let the continuous random variable have a U-shaped distribution with pdf :

defined on a domain of support , where parameter :

Figure 1 plots the parent pdf's when parameter and 5:

Figure 1.

For a sample of size on , the pdf of the order statistic is given by the mathStatica function:

with domain of support:

Figure 2 plots the pdf of as increases from 1 to 10, given a sample size , with :

Figure 2.

The bivariate pdf of two order statistics and , for , is given by: