# Bapat–Beg theorem

Bapat–Beg theorem In probability theory, the Bapat–Beg theorem gives the joint probability distribution of order statistics of independent but not necessarily identically distributed random variables in terms of the cumulative distribution functions of the random variables. Ravindra Bapat and Beg published the theorem in 1989,[1] though they did not offer a proof. A simple proof was offered by Hande in 1994.[2] Often, 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 different statistical population and therefore has its own probability distribution.[1] Contents 1 Statement 1.1 Independent identically distributed case 2 Remarks 3 Complexity 4 References Statement Let {displaystyle X_{1},X_{2},ldots ,X_{n}} be independent real valued random variables with cumulative distribution functions respectively {displaystyle F_{1}(x),F_{2}(x),ldots ,F_{n}(x)} . Write {displaystyle X_{(1)},X_{(2)},ldots ,X_{(n)}} for the order statistics. Then the joint probability distribution of the {displaystyle n_{1},n_{2}ldots ,n_{k}} order statistics (with {displaystyle n_{1}

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