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?


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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