I am trying to obtain CDF in the following expression but not getting it clearly.
$F_{\gamma_l}(\gamma_{th}) = \text{Pr}\biggl(\frac{\zeta|g|^2|k|^2}{\mu_1|g|^2+\mu_2|k^2|+\mu_3|h|^2+\mu_4}\leq \gamma_{th}\biggr)$ ---(1)
where $|g|^2$, $|k|^2$ and $|h|^2$ follows exponential distribution, $\text{Pr}$ denotes probability and all other things are constant.
Also, the random variable of interest is $|g|^2$. Hence, I wrote (1) as
$F_{\gamma_l}(\gamma_{th}) = \text{Pr}\biggl(|g|^2\leq \gamma_{th}\frac{(\mu_2x_k+\mu_3x_h+\mu_4)}{\zeta x_k-\gamma_{th}\mu_1}\biggr)$ ---(2)
where $x_h,x_k$ are variables of integration.
Further, I expressed (2) as
$F_{\gamma_l}(\gamma_{th}) = \int_0^{\infty}\int_0^{\infty} \text{Pr}\biggl(|g|^2\leq \gamma_{th}\frac{(\mu_2x_k+\mu_3x_h+\mu_4)}{\zeta x_k-\gamma_{th}\mu_1}\biggr)\cdot f_{|k|^2}(x_k) \cdot f_{|h|^2}(x_h) $---(3)
My query is did I correctly obtained equation (3) from (2).
Any help in this regard will be highly appreciated.