I have encountered a problem in my work where x,y,z are my independent exponential random variables. I need to find out the probability given I have the limit for random variable z which is $(z \lt \frac{1}{c}) $

$Pr \biggl [\biggl (\frac{\boldsymbol z+a}{(b \cdot \boldsymbol y(1-c \cdot \boldsymbol z))}\leq \boldsymbol x \leq f \boldsymbol , \frac{\boldsymbol z+a}{(d(1-c \cdot \boldsymbol z))}\leq \boldsymbol y\leq \frac{\boldsymbol z+a}{(e(1-c \cdot \boldsymbol z))} \biggr ) \boldsymbol | (z \leq \frac{1}{c}) \biggr ]$

where a,b,c,d,e,f are my constants.

I am trying to find the joint probability of x,y given z. I think I cannot separate them as they are dependent now for my case because of the way the expression is.

I would be really thankful if somebody guide me as how to proceed for such a case as it becomes quite difficult.


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