I need to find the ensemble mean of a quantity containing three random variables $X$, $Y$ and $Z$. Given by: $\frac{X}{Y+Z}$. To find the ensemble mean, I would need the joint probability distribution of these three variables, i.e.,
$P(X,Y,Z)=P(X)*P(Y|X)*P(Z|X,Y)$ which will be utilized to get the ensemble mean of the quantity $\frac{X}{Y+Z}$. I have relationship between $X$ and $Y$. i.e., $Y=y_0-Xa$, where $y_0$ and $a$ are constants, but I don't have a relationship or equation relating $Y$ and $Z$. My first question is: Can I now write $P(X,Y,Z)=P(X)*P(Y|X)*P(Z|X)$ because $Y$ and $Z$ are independent? Second, while calculating $P(Y|X)=\frac{f(y,x)}{f_Y(y)}$, I need to find the joint PDF of Y and X i.e., $f(y,x)$. How to find it? Can I use the formula $f(y,x)=f_Y(y) * f_X(x) * |dx/dy|$?
If I have a particular probability distribution for $X$ and $Y$, e.g, Poisson for $X$ and normal for $Y$, I still want to also make use of the original relationship $Y=y_0-Xa$.