Solving a Challenging Issue Involving Random Variables

[EngWiPy]

Hello,

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

$$\gamma_A^{(1)}\leq\gamma_A^{(2)}\leq\cdots\leq\gamma_A^{(m_A)}\leq\cdots\leq\gamma_A^{(M_A)}$$

and

$$\gamma_B^{(1)}\leq\gamma_B^{(2)}\leq\cdots\leq\gamma_B^{(m_B)}\leq\cdots\leq\gamma_B^{(M_B)}$$

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

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

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

$$\gamma_{\text{eq}}=\frac{\gamma_A^{(m_A)}\,\gamma_B^{(m_B)}}{\gamma_A^{(m_A)}+\gamma_B^{(m_B)}+1}$$

What is the easiest way to find the moment generating function $$\mathcal{M}_{\gamma_{\text{eq}}}(s)= E_{\gamma_{\text{eq}}}\left[\text{e}^{s\,\gamma}\right]$$??

 

[bpet]

where all random variables in the same set are independent and identically distributed random variables

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.