I understand that the p.d.f of order statistics for Independent Non-Identical Distributions are given by the Bapat-Beg theorem as previously explained in another question. As explained in the article, the formula quickly becomes an intractable expression as the number of statistics goes up. I was wondering if the expression simplifies a little bit for the ordered cumulative distribution function. In particular what would it be for the number of non-identical distributions $n=3$ and the order statistics is the middle value $k=2.$
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It seems the correct formula is the following (the three cumulative probability distributions are named $c_1, c_2,\text{ and } c_3$).
$$C_\text{order 2} = c_1 c_2 + c_1 c_3 + c_2 c_3 -2 c_1 c_2 c_3$$