# Probability of three correlated gaussian variables

closed-formnormal distributionprobability

Given three standard gaussian random variable $$X,Y$$ ($$X$$ and $$Y$$ are independent) and $$Z$$ with
$$X,Y = \mathcal{N}(0,1)$$
$$Z = Y-X$$
and $$a,b,c \in \Bbb R$$

Is it possible to calculate analytically this probability $$\Bbb P(X

(by analytically, I mean we can write this probability by using basic mathematical operations and standard normal probability functions $$\Phi_n(\mathbf{z};\mathbf{0}_n, \mathbf{I}_n)$$)

I compute the covariance matrix $$\mathbf{\Sigma}$$ of $$(X,Y,Z)$$, which is
$$\mathbf{\Sigma} = \begin{pmatrix} 1 & 0 & -1\\ 0 & 1 & 1 \\ -1 & 1 & 2 \\ \end{pmatrix}$$

but this covariance matrix is unluckily singular.

Perhaps there is a clever way to decompose the region $$\{X into several regions that can be analytically calculated?

I'll share my progress until now.

For short, we will define $$\Phi(x)=\Phi(x;0,1)$$, that is, the cummulative distribuition function of the Standard Normal and we use $$\phi(x)=\Phi'(x)$$ for the pdf. Using Bayes' Theorem, the independence of $$X$$ and $$Y$$ and the fact that $$Z=Y-X$$ we have

\begin{align*} \mathbb{P}(X

Let's calculate

$$\mathbb{P}(Y

See that $$\mathbb{P}(Y i.e., $$\mathbb{P}(Y

Moreover, define

$$L:=\mathbb{P}(Y

Let's define $$d:=\min\{a,b-c\}$$. Then

\begin{align*} L&=\mathbb{P}(Y

Finally, we have

\begin{align*} \mathbb{P}(X

I don't quite know how to deal with this $$L$$ yet. I'll try to think about that more later. Maybe we can use some idea of this post.