Comparative statistics of (trivariate) random event

[estebanox]

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.

 

[Stephen Tashi]

Specifically, I'm interested in
(∂/∂a)Pr[X>( (Y-a)/Z )]

What's your definition of the relation "$>$" when we are comparing two vectors ?

 

[estebanox]

Each of X, Y and Z are random variables. I refer to random vector when talking about (X,Y,Z)

 

[Stephen Tashi]

Then I don't understand the notation:

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

Does this indicate we are considering 3 different situations for the standard deviations or $3^3$ different situations ?

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 x∈[0,3].

Did you mean "for $a\in [0,3]$" ?

 

[estebanox]

Then I don't understand the notation:Does this indicate we are considering 3 different situations for the standard deviations or $3^3$ different situations ?Did you mean "for $a\in [0,3]$" ?

Oh, yes, typo! I'll fix it. Thanks.

 

[estebanox]

Oh, yes, typo! I'll fix it. Thanks.

The vector μ and the matrix Σ refer to the parameters of the joint distribution of (X,Y,Z). The simulation is fixing all of these parameters, and tracing the probability for different values of "a".

 

[Stephen Tashi]

I suggest we start by seeing if we can get intuition for a much simpler situation!

Let $Y$ and $Z$ each be uniformly distributed on the interval [-1,1]. Let $x$ be a number instead of a random variable. How does $P(x > (Y- a)/Z$ vary with $a$ ?. Maybe we need to consider two cases: $x< 0$ and $x > 0$.

It might be simpler to ask about $P(x < (Y-a)/Z)$ since that's a question about a cumulative distribution.