Please validate my solution (conditional Probability)

Question

I have a conditional probability problem as follows

$\Pr\{x>\gamma,\quad xy+y^2>\gamma\}$

where $x$ and $y$ are i.i.d r.v. After conditioning on r.v. $y$

$=\int_{0}^{\infty} \quad \Pr\{x>\gamma\} \quad\Pr\{x>\frac{\gamma}{z}-z\} \quad f_y(z) \quad dz$

$=\int_{0}^{\infty} \quad (1- F_x(\gamma)) \quad (1-F_x(\frac{\gamma}{z}-z)) \quad f_y(z) \quad dz$

I am not sure if the result in the last equation is correct or not? Am I using the probability conditioning right?

any kind of help will be very much appreciated.

Answer 1

\begin{align} &\Pr\{x>\gamma,\quad xy+y^2>\gamma\} \\ &=\int_{0}^{\infty} \Pr\{x>\gamma, x>\frac{\gamma}{z}-z\} f_y(z) dz \\ &=\int_{0}^\infty Pr(x > \max(\gamma, \frac{\gamma}{z}-z )) f_y(z) \, dz \\ &=\int_0^\infty\left(1- F_x(\max(\gamma, \frac{\gamma}{z}-z)) \right)f_y(z) dz\end{align}