Which equal correlations of three random variables are possible? [duplicate]

3 random variables have equal pairwise correlation, what are the possible values of correlation?

I think 1 and 0 are the obvious answers, but are there any more, and how could I find them?

• Do you know the formula to calculate correlation? May 15 '14 at 20:30
• A more general version of this question is addressed at stats.stackexchange.com/questions/5747.
– whuber
May 15 '14 at 22:09

Let $X_1$, $X_2$, $X_3$ be three random variables with common pairwise correlation coefficient $\rho$, that is $\mbox{corr}(X_i, X_j)= \rho$ for $i \neq j$ with $|\rho|\leq 1$. So, the correlation matrix of $X = (X_1, X_2, X_3)$ is $$\left( \begin{array}{ccc} 1 & \rho & \rho \\ \rho & 1 & \rho \\ \rho & \rho & 1 \end{array} \right) .$$ Correlation matrices need to be positive-semidefinite, which implies that their leading principal minors are all positive. So $\rho$ must satisfy the following two conditions $$\begin{cases} 1 - \rho^2 &\geq 0,\\ 1 - 3\rho^2 + 2\rho^3 &\geq 0 . \end{cases}$$ The first condition is always satisfied, and the second condition implies that $\rho \geq -0.5$.
• In case it's not immediately obvious to everyone $1-3\rho^2+2\rho^3 = (1-\rho)^2(1+2\rho)$ May 16 '14 at 2:59