Assume that $X_i$, $Y_k$, $i=0,\ldots,N$, $k=1,\ldots,K$ are non-negative independent non-identically distributed random variables. Let us define the random variable $Z$ as \begin{align} Z=\frac{aX_0} { \max\limits_{k=1,\ldots,K}Y_k\cdot \left( \sum \limits_{i=1}^{N} b_iX_i +1 \right)} \end{align} where $a$, $b_i$ are non-negative constants. The PDF of $X_i$ and $Y_k$ are defined as $f_{X_i}(x_i)$ and $f_{Y_k}(y_k)$, respectively. The CDF of $X_i$ and $Y_k$ are defined as $F_{X_i}(x_i)$ and $F_{Y_k}(y_k)$, respectively. How could I find the CDF of $Z$?
More specifically, I am going to find the CDF of $Z$ as follows: \begin{align} F_Z(z)=\Pr ( Z < z )=\int\limits_0^\infty \int \limits_0^\infty \Pr \left( \frac{aX_0} { y\left( x + 1 \right)} < z \right) f_Y(y) \; dy \; f_X(x) \; dx \; dy \end{align} where $Y=\max\limits_{k=1,\ldots,K}Y_k$ and $X=\sum \limits_{i=1}^N b_iX_i$, $f_Y(y)$ and $f_X(x)$ are PDFs of $X$ and $Y$, respectively. Note that all random variables are distributed following independent non-identically distributed exponential distributions.
I am not sure whether this formula is correct or not? Could you please verify it for me please?