Hi I have this question.

Suppose that the random variables $\displaystyle X_1, X_2, ... X_n$ are identically distributed, with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$, but not independent. Assume the correlation between any pair is equal to $\displaystyle \rho$. i.e. $\displaystyle corr(X_i,X_j)=\rho$ for $\displaystyle i \neq j$.

1) Derive $\displaystyle Var(\bar{X})$ for this situation.

2) What is $\displaystyle Var(\bar{X})$ when \rho=0? Explain.

3) What is $\displaystyle Var(\bar{X})$ when \rho=1? Explain.

4) Use the rsult you have derived to comment on how small $\displaystyle \rho$ can be in this situation. Explain.

To me, this looks hard. I can do it for the case if the random variables are independent and identically distributed (because it is such a well-known result).

But how do you do it if the random variables are not independent? I'm a bit confused. Can someone lend me a hand and direct me to a suitable source for this question?

Thanks!!

Regards,

Lpd