independent variables or not I have $X$ and $Y$ that are independent random variables, I define $Z_1=X-Y$ and $Z_2=X+Y$. Why $Z_1$ and $Z_2$ are independent as well? Not clear to me at all as I see, say X in both variables, so don't they share a clear "dependency" then between each other? Thanks for any pointers how to show it! 
 A: How about rolling an ordinary six-sided fair die twice; $X$ is value of first roll; $Y$ is value of the second roll.  If you knew, say, that $Z_1=X-Y=5$, does that give you any information about $Z_2=X+Y$? 
If knowing information about one random variable changes your assessment of probabilities for the other random variable, then the variables are not independent.
A: If $X$ and $Y$ are independent then $f_Xf_Y=f_{XY}$.
Consider now $f_{Z_1Z_2}=\frac 12f_{XY}\left(\frac{z_2+z_1}{2},\frac{z_2-z_1}{2}\right) .$ Then $f_{Z_1}=\int_{-\infty}^\infty \frac 12 f_{XY}\left(\frac{z_2+z_1}{2},\frac{z_2-z_1}{2}\right)dz_2$ and similarly $f_{Z_2}=\int_{-\infty}^\infty \frac 12 f_{XY}\left(\frac{z_2+z_1}{2},\frac{z_2-z_1}{2}\right)dz_1$.
Now, $f_{Z_1}f_{Z_2}=\left[\int_{-\infty}^\infty \frac 12 f_{XY}\left(\frac{z_2+z_1}{2},\frac{z_2-z_1}{2}\right)dz_2\right]\left[\int_{-\infty}^\infty \frac 12 f_{XY}\left(\frac{z_2+z_1}{2},\frac{z_2-z_1}{2}\right)dz_1\right]$.
Consider $f_{XY}=4xy\mathcal X_{[0,1]}(x)\mathcal X_{[0,1]}(y)$ and then try to show that $f_{Z_1}f_{Z_2}\neq f_{Z_1Z_2}$.
A: Counterexample:
If $X$ and $Y$ are both integer random variables, then $(X-Y)$ and $(X+Y)$ are simultaneously odd or even. If you knew, say, that $X-Y$ is odd, then $X+Y$ is odd too, so they are not independent.
