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.wikipedia
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Order statistic

order statisticsorderedth-smallest of items
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.
Then they are independent, but not necessarily identically distributed, and their joint probability distribution is given by the Bapat–Beg theorem.

Ravindra Bapat

Bapat, Ravindra
Ravindra Bapat and Beg published the theorem in 1989, though they did not offer a proof.
He is known for the Bapat–Beg theorem.

Permanent (mathematics)

permanentmatrix permanentspermanent of a matrix
is the permanent of the given block matrix.
Bapat–Beg theorem, an application of permanents in order statistics

Probability theory

theory of probabilityprobabilityprobability theorist
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.

Joint probability distribution

joint distributionjoint probabilitymultivariate distribution
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.

Independence (probability theory)

independentstatistically independentindependence
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.

Random variable

random variablesrandom variationrandom
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.

Sample (statistics)

samplesamplesstatistical sample
Often, all elements of the sample are obtained from the same population and thus have the same probability distribution.

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
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.

Statistical population

populationsubpopulationsubpopulations
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.

Block matrix

block diagonal matrixblock diagonalblock matrices
is the permanent of the given block matrix.

Independent and identically distributed random variables

independent and identically distributedi.i.d.iid
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.

Cumulative distribution function

distribution functionCDFcumulative probability distribution function
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.