Problem: I'm interested in studying the probability of an event involving a random vector.
Specifically, I'm interested in
$$\frac{\partial}{\partial a}Pr\left(X > \frac{Y-a}{Z} \right)$$
where $a$ is a non-random parameter, and the vector $(X, Y, Z) \sim$ Normal $(\mu, \Sigma)$ for $\mu=(0,0,0)$ and $\Sigma=\left( \begin{array}{ccc} 1 & 0.5 & 0.5 \\ 0.5 & 1 & 0 \\ 0.5 & 0 & 1 \\ \end{array} \right)$
What I have:
Simulations tell me that the above partial derivative is non-monotonic, i.e. increasing the value of $a$ first increases the probability of the event, and then decreases it. Here is a simulation for $a\in[0,3].$
I can see that this is likely coming from the fact that $Z$ takes value along the real line, so the effect of $a$ flips with the sign of $Z$.
Question I struggle to understand exactly what is going on here from a distributional point of view. What determines the point of inflection? I don't need an analytical solution – just some intuition.