0

Two density functions F and G

Can I make any statement about the covariance from just looking at these two density functions? My intuition is the following: How could the covariance be high here if the probability for high values is low for F and high for G. If they were correlated the shape would have to be similar right?

user2820379
  • 313
  • 1
  • 2
  • 11

1 Answers1

0

Let the two functions cross each other at $a$. Then let $X\sim F $ and let

$$Y=a+a-X=2a-X.$$

Now,

$$\mathbb F_Y(y)=P(Y<y)=P(2a-X<y)=P(X>y-2a)=$$ $$=1-P(X\le 2a-y)=1-\mathbb F_X(2a-y).$$

For the densities, then, we have

$$f_Y(y)=f_X(2a-y).$$

See the following example now:

enter image description here

In this case the two random variables determine each other and the densities are like you imagined. That, is the relationship between the shape of the densities has nothing to do with correlation.

zoli
  • 20,452
  • Nice answer. Took me a while to understand because I'm not used to think in these ways. I conclude that you can not say anything about the covariance just from knowing univariate density functions. However, I think if you know the joint bivariate distribution over X and Y then you could calculate the covariance. – user2820379 Aug 09 '15 at 18:56
  • @user2820379: Right. And thank you. – zoli Aug 09 '15 at 19:21