Dependence of certain random variables Consider $X_1,X_2$ i.i.d. standard normal random variables(mean 0, variance 1). Are the random variables $Y=X_1+X_2$ and $Z=X_1-X_2$ dependent? I am not sure how to prove this one way or the other.

If you are familiar with the concept of characteristic function, it is easiest to compute characteristic function for $(Y, Z)$. For independent variables, the characteristic function would factor into a product:
$$
 \begin{eqnarray}
  \mathbb{E}\left( \exp( i t_1 Y + i t_2 Z ) \right) &=& 
  \mathbb{E}\left( \exp( i (t_1+t_2) X_1 + i (t_1-t_2) X_2 ) \right) \\ & = & 
  \exp\left( -\frac{1}{2} \left(t_1+t_2\right)^2 \right)  \cdot 
  \exp\left( -\frac{1}{2} \left(t_1-t_2\right)^2 \right)  \\
  &=& \exp\left( -t_1^2 \right) \cdot \exp \left(-t_2^2 \right)
 \end{eqnarray}
$$
Hence the $Y$ and $Z$ are independent normal with mean 0 and standard deviation of $\sqrt{2}$.
