Let me follow up on the theme – talked about here – of distributing an operator over another. To recap, we saw two ways of distributing an expectation
Can we say anything like this about variance? Consider the variance of the sum of random variables (recall that ).
Looks like we have an extra term. Before trying to see what this is, can we think of a situation when this term could be and thus give us ? The answer is yes because if and are independent, then .
In general, though, this extra term will be non-zero. How can we make sense of this extra term? If you go back to the Cauchy-Bunyakovskii inequality we looked at, we can write this term as
So it looks as if we can relate the extra term to the variance of the random variables. More specifically, we see that
And this gets to be called the correlation coefficient. Why is this interesting you ask? First, note that
You’ve often heard how a lack of correlation does not imply independence of random variables. Well, the above identity is the conclusive proof. Because
Will furnish with an example of when this happens. It seems tricky to come up with an example.