Beyond simple addition: The nuances of V(X+Y) and covariance

Why adding only individual variances can lead to underestimation

In statistics and probability, what is the variance of the sum of 2 random variables?

Expressing my question in mathematical notation, what is

\(V(X + Y)?\)

It is very tempting to conclude that,

\(V(X + Y) = V(X) + V(Y),\)

but this is true ONLY if the random variables X and Y are uncorrelated. (An even stronger condition is independence, which is what many practitioners think about in their real-life work.)

If X and Y are positively correlated, then simply adding V(X) and V(Y) will underestimate the variance of their sum, because it ignores the covariance between X and Y. This can lead to disastrous consequences in all types of applications, such as making business investments, performing medical treatments, or building public infrastructure.

If X and Y are correlated, then

\(V(X + Y) = V(X) + V(Y) + 2\text{Cov}(X, Y).\)

Notice the extra term of the covariance multiplied by 2. It is also possible to overestimate V(𝘟 + 𝘠), because the covariance can be negative.

More generally, if you have scalar constants multiplying the random variables, then the ensuing variance of the sum is

\(V(aX + bY) = a^2V(X) + b^2V(Y) + 2ab\text{Cov}(X, Y)\)

This is a common mistake for students who first learn mathematical statistics, so make sure to remember that last term of 2Cov(X, Y)!