If X, Y independent then X/Y and Y are independent?

Question

if two random Variables $X, Y$ are independent

Does that mean that $\left(\frac{\ X }{Y}\right)$ and $Y$ are independent?

for example is it true that E[$\left(\frac{\ X }{Y}\right)$|$Y$]= E[$\left(\frac{\ X}{Y}\right)$] because of the independence of the random variables X and Y?

Answer 1

Suppose that $X,Y$, satisfy $\mathbb{P}( X=-1 ) =\mathbb{P}( X=1 )=1/2 $ and $\mathbb{P}( Y=-1 ) =\mathbb{P}( Y=2 )=1/2 $.

Set $Z=X/Y$, then $\mathbb{P}( Z=1/2, Y=-1)=0$ however $\mathbb{P}(Z=1/2)\neq 0$ and $\mathbb{P}( Y=-1)\neq 0$.

So, they are not independent.

For the conditional what we have is $E\left ( X/Y\right |Y=y ) = E(X/y)$