[R] covariance question which has nothing to do with R

[Leeds, Mark (IED)]

This is a covariance calculation question so nothing to do with R but
maybe someone could help me anyway.

Suppose, I have two random variables X and Y whose means are both known
to be zero and I want to get an estimate of their covariance.

I have n sample pairs 

(X1,Y1)
(X2,Y2)
.
.
.
.
.
(Xn,Yn)

, so that the covariance estimate is clearly 1/n *(sum from i = 1 to n
of ( X_i*Y_i) ) 

But, suppose that it is know that the X_i are positively correlated with
each other and that the Y_i are independent
of each other.

Then, does this change the formula for the covariance estimate at all ?
Intuitively, I would think that, if the X_i's are positively
correlated , then something should change because there is less info
there than if they were independent but i'm not sure what should change
and I couldn't find it in a book.  

I can assume that the correlation between the X_i's is rho if this makes
things easier ? Thanks.

References are appreciated also.
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