I have been doing exercises in basic probabilities, and there is a type of questions which I am never quite sure of the answer, so I wanted to get some opinion about the way I do it.
The question is the following: Consider $W = XY - Z$, where $\{X,Y,Z\}$ follows a Normal $(\mu,\Sigma)$ and I want to get $\Pr(W>0)$.
In the case where $W$ would be a transformation of two variables it's not that hard to figure out what the domain of integration on a plane is, but here can I proceed in the following way? $$ \Pr(W>0\mid X=x) = \Pr(Z<XY\mid X=x) \\ = \int_{-\infty}^{+\infty} \int_\infty^{xy} \frac{\phi(x,y,z)}{\phi_X(x)} \, dz \, dy $$ And then use the law of total probability to get: $$ \Pr(W>0) = \int_{-\infty}^{+\infty} \Pr(W>0\mid X=x) \phi_X(x) \, dx $$
Thank you