Bapat–Beg theorem

In probability theory, the Bapat–Beg theoremR. B. Bapat and M. I. Beg. Order statistics for nonidentically distributed variables and permanents. Sankhyā Ser. A, 51(1):79–93, 1989. gives the joint cumulative distribution function of order statistics of independent but not necessarily identically distributed random variables in terms of the cumulative distribution functions of the random variables. A simple proof of this can be found in Sayaji Hande. A note on order statistics for nonidentically distributed variables Sankhyā Ser. A, 56(2):365–368, 1994.

Ordinarily, all elements of the sample are obtained from the same population and thus have the same probability distribution. The Bapat–Beg theorem describes the order statistics when each element of the sample is obtained from a possibly different population with a different probability distribution.

The theorem

Let <math>textstyle X_</math>, <math>textstyle i=1,ldots,m</math> be independent real valued random variables with cumulative distribution functions <math>textstyle F_left( xright) </math>. Denote the order statistics by <math>textstyle Y_,Y_,ldots,Y_</math>, with <math>textstyle Y_leq Y_leqcdotsleq Y_</math>. Further let <math>textstyle y_=-infty,</math> <math>textstyle...
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