Suppose I have three continuous random variables $X, Y, Z$, and all of them are correlated $( {\rho_{x,y}, \rho_{x, z}, \rho_{y, z})}$. And I'm interested in the joint CDF: $P(X<0, Y<0, Z<0)$. Which is a function of $(\rho_{x,y}, \rho_{x, z}, \rho_{y, z}, X,Y,Z)$
Can I create a super random variable $A := (X, Y)$ to replace $(X, Y, Z)$ with $(A, Z)$ such that $P(A<0)=P(X<0, Y<0)$? And somehow obtain $\rho_{A, Z}$ so that $P(X<0, Y<0, Z<0)$ can be simplified into only $P(A<0, Z<0)$.
I am hoping to extend this to high dimension, Given $(X_1, X_2, X_3, X_4)$, one can reduce the joint distribution by $B := X_2, X_3$, into $(X_1, B, X_4)$. And doing further reduction via $C := (B, X_4)$ to have a bivariate distribution $(X_1, C)$
Probably it's a bad idea to doing so, but I can't figure out why it is wrong?
Say I have two random variables $X$ and $Y$ but two random variables is a lot so I'm going to define $W = (X,Y)$. How do I graph $W$? In two dimensions - one for $X$ and one for $Y$ - otherwise I've lost information. Thus it was a pointless endeavor.