Is the dependency between 2 random variables always symmetric?

Question

If you have 2 random variables, $X$ and $Y$, is the dependency symmetric? In other words, if $Y$ depends on $X$, is $X$ dependent on $Y$? If not, what is a counter example?

Answer 1

Yes. Standard textbooks usually define independence between pairs of random variables, and less of it as a "directed" relationship.

It might be helpful look at it using conditional probability. (I'm using discrete random variables for simplicity.) For any two r.v.s $X$ and $Y$,

$p(X,Y) = p(X) p(Y|X) = p(Y)p(X|Y)$

Here, you can say that $Y$ depends on $X$ if $p(Y|X) \ne p(Y)$, and vice versa. Note that if $p(Y|X) \ne p(Y)$, then $p(X|Y)$ cannot be equal to $p(X)$. (Consider what happens otherwise.) In this sense, if $Y$ depends on $X$, then $X$ depends on $Y$. The converse is also true.