Can I make statements about the covariance from looking at univariate distributions?

Question

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?

Answer 1

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:

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.