Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question

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.

share|cite|improve this answer

This type of questions is often asked on job interviews, so I share a full solution of this more general form.

Q: We have 3 variables, $X, Y, Z$. In what range can be the correlation between them?

A: We can put the 3 variables in a correlation matrix \begin{equation*} \left( \begin{array}{c c c} 1 & \rho(X, Y) & \rho(X, Z) \\ \rho(Y, X) & 1 & \rho(Y, Z) \\ \rho(Z, X) & \rho(Z, Y) & 1 \end{array} \right) \end{equation*} which must be positive and semi-definite; that means that its determinant must be non-negative. So, they must satisfy \begin{equation*} 1 + 2 \rho (X, Y) \rho (X, Z) \rho (Y, Z) - \rho(X, Y)^2 - \rho(X, Z)^2 - \rho(Y, Z)^2 \geq 0; \end{equation*} or, for given $\rho (X, Y); \rho (X, Z);$ thinking about this as the roots of a polynomial, we see that we must have $\rho(Y, Z)$ inside the open interval \begin{align*} ( \rho(X, Y) \rho(X, Z) &- \sqrt{ \rho(X, Y)^2 \rho(X, Z)^2 + 1 - \rho(X, Y)^2 - \rho(X, Z)^2}; \\ ~~~~~~~~ \rho(X, Y) \rho(X, Z) &+ \sqrt{ \rho(X, Y)^2 \rho(X, Z)^2 + 1 - \rho(X, Y)^2 - \rho(X, Z)^2} ). \end{align*} In practice, this means that if, for example, $\rho(X, Y) = \rho(X, Z) = 1$; for $\rho(Y, Z)$, the value must be 1. This can also be used to find answer to a common interview question: If $\rho(X, Y)= 0.9$, $\rho(X, Z) = 0.8$, Can $\rho(Y, Z) = 0.1$? Here, we see that it is not possible.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.