How to calculate the probability $\Pr \left[ {X > 2Y,X > Z + \frac{1}{Y}} \right]$ where X,Y,Z are independent exponential random variables?

Question

In my research where I need to calculate the successfully decocode probability $\Pr \left[ {X > 2Y,X > 3Z + \frac{1}{Y}} \right]$ where X,Y,Z are independent exponential random variables with parameters ${\lambda _1},{\lambda _2},{\lambda _3}$.

From my first glance, I want to calculate these events separately but due to the share variable $X$ I feel that I am wrong.

In which way should we approach a problem like this ? Is is possible to split the probability using conditional independence if we condition on the variable $X$

Please help me with this problem

Thank you for your enthusiasm !

Answer 1

Hint 1: Suppose $X, U$ are independent random variables in $\mathbb{R}$, then $P(X > U) = \int_{-\infty}^{+\infty} P(X > x)p_U(x)dx$, where $p_U(x)$ is the probability density function of $U$. This formula can be used in your problem.

Hint 2: Take $U = \max(2Y, 3Z + \frac{1}{Y})$