Why does a negative covariance reduce the variance of the sum of two dependent variables?

Question

If I am interested in Var(X + Y)=Var(X) + Var(Y) + 2Cov(X,Y) where X and Y are dependent iid random variables, there is the possibility that the covariance could be negative, which would yield a smaller V(X + Y) than if the covariance was positive. Intuitively, I do not understand why that is. If they are positively related, wouldn't X and Y be more likely to be close to each other, and hence the variance would be smaller?

Answer 1

You're asking for the variance for $X+Y$, and not the difference between $X$ and $Y$. It's a measure of deviation from the mean of $X+Y$.

When the variables are positively correlated, each random variable has it's own deviation and they tend to add up. (e.g. $X$ goes to +5 and $Y$ goes to +5) Thus, the variance for $X+Y$ is higher.

When the variables are negatively correlated, the deviation tend to "cancel" (e.g. $X$ goes to +5 and $Y$ goes to -5), thus the variance for $X+Y$ is smaller.

Answer 2

Variance is a measure of to what degree samples will deviate from the mean, not a measure of where the mean is. So it's not a measure of $X-Y$ or $X+Y$ as you imply, it's a measure of the difference between one $X+Y$ and another.

Answer 3

Let's try an extreme case: $$ (X,Y) = \begin{cases} (1,4) \\ (2,3) \\ (3,2) \\ (4,1) \end{cases} \text{ each with probability } 1/4. $$ Then $\operatorname{var}(X+Y)=0,$ whereas $\operatorname{var}(X) = \operatorname{var}(Y)>0.$

Answer 4

Consider $X$ as the count of heads among ten coin tosses, and $Y$ as the count of tails among the same.   These variables are clearly negatively correlated, and the variance of their sum is zero.   The sum is, after all, certainly ten.

For an intuitive feel, you might consider that the wandering of sample values for negatively correlated random variables from their means will tend to be in opposite directions, thus reducing the wandering of their sum.   Where as the wandering of sample values for positive correlated random variables will tend to be in the same direction, reinforcing the wandering of their sum.

Thus as in the example, whenever you flip the coins to find the count of heads to be greater than five, then the count of tails will be less than five, and vice versa.   The value of their sum, however, won't be very far from ten at all.