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A Challenging Issue

  1. Sep 19, 2009 #1

    Suppose that we have two sets of random variables, which are arranged in an ascending order as:




    where all random variables in the same set are independent and identically distributed random variables, which are characterized as central Chi-square with [tex]2\,N_i[/tex] degrees of freedom, i.e.:


    for [tex]i\in\{A,\,B\}[/tex]. Now suupose that a new random variable is formed as following:


    What is the easiest way to find the moment generating function [tex]\mathcal{M}_{\gamma_{\text{eq}}}(s)= E_{\gamma_{\text{eq}}}\left[\text{e}^{s\,\gamma}\right][/tex]??
  2. jcsd
  3. Sep 25, 2009 #2
    Assuming you mean they are the order statistics of set of independent random variables - the Wikipedia article has some distribution formulas. From there the mgf could be expressed as a 2D integral.
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