How do I find the pdf and cdf of the j-th order statistic?

Suppose the number of observations associated with rank j is

nj , 1 ≤ j ≤ n, and the corresponding proportion is

denoted as

qj = nj/N

, where N = Summation( nj )

For each rank j, the observations xi(j), i= 1, ..., nj ,

are independent, which can be taken as realizations of the j-th order statistics.

The PDF and CDF of the j-th order statistics are given by fj (·) and Fj (·), respectively, where

fj (x) = g(F(x); j, n + 1 − j),

Fj (x) = G(F(x); j, n + 1 − j),

where F(·) is the CDF of the population from which samples are being taken, and g(·) and

G(·) are the PDF and CDF of Beta distribution, respectively.