General question: What is the distribution for the maximum of 2 independent draws from cdf F(x), when we know that the minimum of those same two draws is the kth order statistic of the minimum of n pairs of independent draws from F(x)? Less technically, what is the distribution of the maximum associated with the kth greatest (of n) minima? 


A specific example:
Assume 8 independent draws from cdf F(x), which is defined over 0 to 1. Then, arbitrarily group the draws into 4 pairs. Compare the minimums of each pair. Label the maximum of these minimums as “a”. Label a’s pair (which is by definition > a) as “b”. Now, choose among the other three pairs arbitrarily, and label the two values in that pair as “c” and “d” (where c is the min of the pair and d is the max of the pair).


What are the distributions of b and d? 


I know the distribution of a: F(a) = (1-(1-F(x))^2)^4 =Max of 4 draws of the Min of 2 draws of F(x).


I also know the distribution of c: F(c) = mixture of 1st , 2nd, and 3rd order statistics of 4 draws of Min of 2 draws of F(x). I get this by averaging the integrals (wrt x) for the pdfs that result from substituting (k=1, n=4), (k=2,n=4) and (k=3, n=4) into the following equation:
(n!/((k - 1)!*(n - k)!))*(F(x)^(k - 1))\*((1 - F(x))^(n - k))\*F'(x)


I don’t know how to define F(b) or F(d)


And help would be greatly appreciated. 



Probability distributions:  The maximum of a pair of iid draws, where the minimum is an order statistic of other minimums?
This question has been answered by Bogdan Lataianu at this link:


<https://stats.stackexchange.com/questions/13259/what-is-the-distribution-of-maximum-of-a-pair-of-iid-draws-where-the-minimum-is>


