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A Challenging Issue

  1. Sep 19, 2009 #1
    Hello,

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

    [tex]\gamma_A^{(1)}\leq\gamma_A^{(2)}\leq\cdots\leq\gamma_A^{(m_A)}\leq\cdots\leq\gamma_A^{(M_A)}[/tex]

    and

    [tex]\gamma_B^{(1)}\leq\gamma_B^{(2)}\leq\cdots\leq\gamma_B^{(m_B)}\leq\cdots\leq\gamma_B^{(M_B)}[/tex]

    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.:

    [tex]f_{\gamma_i}(\gamma)=\frac{\gamma^{N_i-1}}{\overline{\gamma}_i^{N_i}(N_i-1)!}\text{e}^{-\gamma/\overline{\gamma_i}}[/tex]

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

    [tex]\gamma_{\text{eq}}=\frac{\gamma_A^{(m_A)}\,\gamma_B^{(m_B)}}{\gamma_A^{(m_A)}+\gamma_B^{(m_B)}+1}[/tex]

    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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