1.6 Moment Generating Functions and Independence

If $X$ and $Y$ are random variables with the same variance, prove that $\operatorname {Cov}\left[ X+Y, X-Y \right] = 0$. Give a counterexample which shows that zero covariance does not necessarily imply independence.

Let $X$ and $Y$ be discrete random variables taking values 0 or 1 only, and let $\operatorname {pr}(X = i, Y = j) = p_{i j}(i = 1,0 ; j = 1,0)$. Prove that $X$ and $Y$ are independent if and only if $\operatorname {Cov}\left[ X, Y \right] = 0$.

If $X$ is a random variable with a density function symmetric about zero and having zero mean, prove that $\operatorname {Cov}\left[ X, X^{2} \right] = 0$.

If $X, Y$ and $Z$ have joint density function

prove that they are pairwise independent but not mutually independent.