Correlations between 3 random variables

Question

I have a question as follows:

The correlation coefficients between three random variables are x, y, and z respectively. What relation do x, y, and z have to satisfy?

Can someone help me? Thanks very much in advance!

Answer 1

Suppose the correlations among the three pairs are found using the same data points, then a relation can be found. For variables a and b, the correlation is basically just the cos of the angle between them in n dimensional space (n = number of points in the data set), being the dot product of unit vectors along them.

The differences between the unit vectors will follow the triangle inequality. If the angle between unit vectors a and b is $\theta$, then $||a-b|| = 2|sin(\frac{\theta}{2})| = \sqrt{2-2cos\theta}$

So we can say that $\sqrt{2-2x} \le \sqrt{2-2y} + \sqrt{2-2z}$

So $\sqrt{1-x} \le \sqrt{1-y} + \sqrt{1-z}$

So ${1-x} \le {1-y} + {1-z} + 2\sqrt{(1-y)(1-z)}$

Which gives $x \ge y + z - 1 - 2\sqrt{(1-y)(1-z)}$

Of course, similar inequalities hold if we permute x, y, z in any order.