If X and Y are independent random variables such that X = A + B and Y = C + D? Are the pairs (A, C), (B, C), (A, D), (B, D) also independent? By this I mean whether A and C are independent, B and C are independent and so on.
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1$\begingroup$ Let's see: $(A,C)-(B,C)-(A,D)+(B,D)=(0,0)$ shows those four pairs are linearly dependent. Can a set of linearly dependent random variables be independent? $\endgroup$– whuber ♦Mar 26, 2014 at 17:21
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1$\begingroup$ @whuber: I mean whether A and C are independent, B and C are independent and so on. $\endgroup$– user42598Mar 26, 2014 at 17:31
1 Answer
Without any further information about $A, B, C, D$, it is not possible to say anything about their independence. For example, suppose that
$$\begin{align} A &= \frac{X+Y}{2}\\ B &= \frac{X-Y}{2}\\ C & = \frac{X+Y}{2} = A\\ D &= \frac{Y-X}{2} = -B \end{align}$$
Then, $A$ and $C$ cannot be independent since they are identical and neither can $B$ and $D$ be independent since their sum is $0$. In general, $B$ and $C$ are not independent, but in special cases, they might be. For example, if $X$ and $Y$ are independent normal random variables with identical variance, then $\frac{X+Y}{2}$ and $\frac{X-Y}{2}$ are independent. In this special case, the pairs $(A,B)$, $(A, D)$, $(B, C)$, and $(C,D)$ are independent while $(A,C)$ and $(B,D)$ are not. Other choices of $A, B, C, D$ will give different results. So, as stated earlier, there is not anything concrete that can be said in the absence of any information about $A,B, C, D$.