How to calculate the correlation coefficient from minimal distributional assumptions? Let random variables  $X_1,X_2,\ldots,X_n$ satisfy $$(X_i,X_j)\stackrel{d}{=}(X_1,X_2)\quad \forall i, j$$
(that is, these variables are identically distributed and all their bivariate marginal distributions are the same, too) and suppose $$\sum_{i=1}^n X_i=0.$$
How can the correlation coefficient of $X_1, X_2$ be calculated?
 A: The correlation coefficient $\rho$ will exist provided the $X_i$ have a finite nonzero variance (which easily implies $n \ge 2$ because in case $n=1,$ $X_1=0$ has zero variance).  Since $X_1$ and $X_2$ have the same distribution their variances are equal, too, whence
$$\rho = \frac{\operatorname{Cov}(X_1,X_2)}{\sqrt{\operatorname{Var}(X_1)}\sqrt{\operatorname{Var}(X_2)}} =  \frac{\operatorname{Cov}(X_1,X_2)}{\operatorname{Var}(X_1)}.\tag{*}$$
Take the one numerical fact you have--namely, the $X_i$ sum to zero--and compute a variance, using the assumption of identical joint distributions to replace all occurrences of $\operatorname{Var}(X_i)$ by $\operatorname{Var}(X_1)$ and $\operatorname{Cov}(X_i, X_j)$ by $\operatorname{Cov}(X_1, X_2):$
$$0 = \operatorname{Var}(0) = \operatorname{Var}(X_1 + \cdots + X_n)
= n \operatorname{Var}(X_1) + n(n-1)\operatorname{Cov}(X_1,X_2).$$
Dividing both sides by $n\operatorname{Var}(X_1)$ and substituting $(*)$ yields the equation
$$0 = 1 + (n-1)\rho$$
whose solution is easily found.
