Let X$_1$,X$_2$ be two independent random variables on the same probability space (Ω,$\mathcal{F}$,$\mathbb{P}$) that take values 1 and 0 with probability equal to 1/2. Calculate the covariance of X$_1$ + X$_2$ and |X$_1$ - X$_2$|. Are these two random variables independent? Justify your answer
I have written an explanation for the linearity properties of the covariance but wanted to check the calculation for the covariance and need some help determining whether the variables are independent.
For clarification, I believe the question is trying to ask whether the covariance of $X_1$ + $X_2$ is independent to the covariance of |$X_1$ - $X_2$|, rather than whether $X_1$ and $X_2$ are independent.
You are told that $X_1$ takes on values 0 and 1 with probability 1/2 each. Nothing is said about $X_2$! That is, for $X_2$, $P(0)= \frac{1}{2}$ and $P(1)= \frac{1}{2}$, irrespective of what $X_2$. That is precisely what "independent" means!