- #1

estebanox

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**Problem:**I'm interested in studying the probability of an event involving a random vector.

Specifically, I'm interested in

(∂/∂a)Pr[X>( (Y-a)/Z )]

Where "a" is a non-random parameter and the random vector {X,Y,Z} is distributed Normal( µ, Σ)

for µ={0,0,0}

and Σ= {{1, 0.5, 0.5}, {0.5, 1, 0}, {0.5, 0, 1}}

**What I know:**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. Attached is a simulation for a∈[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.

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