I have a question about Didier Piau's answer: Why does it follow that "the correlation of $X_1$ and $X_2$ is $−1/3$ and not $−1/2$ as it should be"?
Perhaps I misunderstood the setting, but if $X$ is uniformly distributed over the points $(1,0,0)$, $(0, 1, 0)$ and $(0,0,1)$, then $E(X_i) = 1/3$, $E(X_iX_j)=0$, and hence $Var(X_i) = 2/9$ and $Cov(X_iX_j) = -1/9$. Therefore, the correlation between $X_i$ and $X_j$ is $(-1/9)/(2/9) = -1/2$, as it should be. In fact I think that, in this setting, the correlation matrix of $X_1, X_2, X_3$ is exactly the matrix $C_3$.
