Covariance and correlation coefficient

[Josh S Thompson]

How do you prove that the maximum value of 2*cov(x,y) can be is equal to var(x) + var(y).

Moreover, how do you prove that the correlation coefficient, cov(x,y)/(sigma(x)*sigma(y), can only be between -1 and 1.

 

[Josh S Thompson]

Forget the first sentence the question is only the second sentence

 

[Dr. Courtney]

Try computing a bunch for random and real data sets and you will see that the rule is never violated.

Of course, I'm an experimentalist.

 

[Josh S Thompson]

is there a proof?

 

[Dr. Courtney]

is there a proof?

I bet there is for a statement for which no counter example exists. I think I recall even seeing one when I taught statistics.

But you can probably Google it up as easily as I can.

 

[Josh S Thompson]

Google sucks, I want some pictures bro. Because I did some examples and I don't understand, I think it doesn't violate those rules because of like dot products or something but I don't see the correlation coefficient. Can someone please enlighten me with some insight.

 

[mathman]

It is a special case of the Schwarz inequality. Essentially $|E(XY)|^2\le E(X^2)E(Y^2)$, where X and Y are the random variables centered at their means.

https://en.wikipedia.org/wiki/Cauchy–Schwarz_inequality