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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]??

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