# A Comparative statistics of (trivariate) random event

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1. Feb 10, 2017

### 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.

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Last edited: Feb 10, 2017
2. Feb 10, 2017

### Stephen Tashi

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

3. Feb 10, 2017

### estebanox

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

4. Feb 10, 2017

### Stephen Tashi

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 $a\in [0,3]$" ?

5. Feb 10, 2017

### estebanox

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

6. Feb 10, 2017

### estebanox

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".

7. Feb 10, 2017

### 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.