My questions seems pretty simple but I am really confuses with following statement Considering w,x,y and z as exponential independent random variables with ZERO mean. What will be probability of following
Pr(wx< Ayz+B)
where A and B are constants. I tried with following integration limits but it is not giving the correct results.
$1- \displaystyle\left(\int_0^{\infty} \int_0^{\infty}\int_0^{\infty} e^{\dfrac{-yzA}{w}} e^{-y}e^{-z}e^{-w} \,dy\,dz\,dw\right) \left(\int_0^{\infty}e^{-\dfrac{B}{w}-w} \,dw\right)$
It's odd.. you mention "trying integration limits" but you don't have any non-trivial limits on your integrals.
This is in the form $P(Y < aX + b)$ where $Y$ and $X$ are independent random variables with the distribution of the product of two standard exponentials (I assume that the four exponentials are independent). If $f$ is the PDF for the product of two exponentials, then you have $$ P(Y< aX+b) = \int_0^\infty dx\int_0^{\max(ax+b,0)}dy f(x)f(y)$$ where I took care of the fact that $X$ and $Y$ are both positive. This is just integrating the joint PDF $f(x)f(y)$ over the region where $y < ax + b.$
Then you need to find $f$, the PDF for the product of two independent exponentials.