Let $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$ be random variables defined as:

\begin{align} \Gamma_1 &= \frac{XY}{X + Y + 1} \\[10pt] \Gamma_2 &= \frac{XY}{aX + bY + c} \\[10pt] \Gamma_3 &= \frac{XY}{aX + bY + cZ + d} \end{align}

where $X$, $Y$, and $Z$ follow exponential distributions with different parameters, and $a$, $b$, $c$, and $d$ are positive constants.

Even though I have the exact CDF for $\Gamma_1$ and $\Gamma_2$, it becomes mathematically intractable to derive the exact CDF of $\Gamma_3$. So I was thinking of trying to find either the upper bound or the lower bound.

Question: How can I evaluate the bounds for $\Gamma_3$?

Should I try to evaluate the lower bound or the upper bound?


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