# Independence of sums and products of independent random variables

Let $x_1,x_2,x_3$ be (real and continuous) random variables.

1. If $x_1,x_2,x_3$ are mutually independent, then $x_1 + x_2$ and $x_2 + x_3$ are independent?
2. If $x_1,x_2,x_3$ are mutually independent, then $x_1 x_2$ and $x_2 x_3$ are independent?
3. What If $x_1,x_2,x_3$ are pairwise independent?

In general, when I prove things like these, what kind of technique do I need to use?

• The answer for all three questions is: not nesessarily. In fact that the asserted independence is false except in very special cases. For example, in 1) x1+x2 and x2+x3 are independent only when x2 is a constant random variable. The best way to handle questions like these is to use characteristic functions. – Kavi Rama Murthy Jan 20 '17 at 6:08

(1) False. Let $X_1$, $X_2$, $X_3$ be mutually independent normal$(0,1)$ variables. Compute $$E(X_1+X_2)(X_2+X_3) = E(X_1X_2 + X_1X_3+X_2^2 + X_2X_3) = 0 + 0 + 1 + 0=1.$$ This is different from $$E(X_1+X_2)E(X_2+X_3)=0,$$ and therefore $X_1+X_2$ and $X_2+X_3$ are not independent.
(2) False. Take the $X$'s from (1) and define $Y_i := \exp(X_i)$ . Then the $Y$'s are mutually independent. But $Y_1Y_2$ and $Y_2Y_3$ are not independent; if they were, then $\log(Y_1Y_2):=X_1+X_2$ and $\log(Y_2Y_3):=X_2+X_3$ would be independent.
(3) False. Find normal$(0,1)$ variables $X_1$, $X_2$, $X_3$ that are pairwise independent. For example, see two of the answers to this question. Next, note that the calculation in (1) requires only pairwise independence.