pairwise correlation of three random variables Assume three random variables have all equal pairwise correlation. What are the possible values of this correlation? Can all of these values be achieved?
The solution says $\rho \in [-\frac 12,1]$, but without any explanation. Can someone give me a hint?
 A: Hint: There exist random variables $X_1$, $X_2$ and $X_3$ with pairwise correlations $\rho_{12}$, $\rho_{13}$ and $\rho_{13}$ if and only if the matrix
$$\Sigma = \begin{pmatrix}
1 & \rho_{12} & \rho_{13}\\
\rho_{12} & 1 &\rho_{23} \\
\rho_{13} & \rho_{23} & 1
\end{pmatrix}
$$
is positive semidefinite. We get that there exist random variables $X_1$, $X_2$ and $X_3$ with all pairwise correlations equal to to $\rho$ if and only if the matrix
$$\begin{pmatrix}
1 & \rho & \rho\\
\rho & 1 &\rho \\
\rho & \rho & 1
\end{pmatrix}
$$
is positive semidefinite (see below for the explanation). This matrix is positive semidefinite precisely when $\rho\in[-1/2,1]$.
Note that there exist $d$ random variables $X_1,\dots,X_d$ s.t. $\mathrm{cor}(X_i,X_j) = \rho$ (for $i\neq j$) if and only if $\rho\in [-\frac{1}{d-1},1]$.

Here is a brief explanation why there exist random variables $X_1$, $X_2$ and $X_3$ with pairwise correlations $\rho_{12}$, $\rho_{13}$ and $\rho_{13}$ if and only if $\Sigma$ is positive semidefinite.
Suppose that the matrix
$\Sigma$ is positive semidefinite. Let 
$$\Gamma = \begin{pmatrix}\gamma_1 \\ \gamma_2 \\ \gamma_3\end{pmatrix} \sim {\cal N}(0,\Sigma)$$
be the multivariate normal random variable with covariance matrix $\Sigma$. Then the correlation of $\gamma_i$ and $\gamma_j$ equals $\rho_{ij}$.
Now suppose that $X_1$, $X_2$ and $X_3$ has correlations $\rho_{ij}$. We may assume that ${\mathbb E}[X_i] = 0$ and $\mathrm{cov}[X_i]=1$. Then the covariance matrix of $(X_1,X_2,X_3)$ is 
$$\Sigma = \begin{pmatrix}
1 & \rho_{12} & \rho_{13}\\
\rho_{12} & 1 &\rho_{23} \\
\rho_{13} & \rho_{23} & 1
\end{pmatrix}.
$$
Therefore, $\Sigma$ is positive semidefinite.
A: There is another interpretation of the problem: you may approach it geometrically. It will be much easier this way. It is a well-known fact, that you can take correlations as cosines. And random variables are vectors in 3D space. Then the question is just how to construct three vectors with same pairwise angles between them. What angles can it be? 
There is a theorem, which is not so hard to prove, that such angles cannot exceed 120 degrees. So, then cos ie correlation is greater, than -1/2. 
You can read more about geometrical interpretation of probability in the book The Geometry of
Multivariate Statistics by Thomas Wickens
