Combination of 2 random variables that is perfectly correlated with another random variable

I'd appreciate some pointers for the following question: Given 3 random variables $X_i$, $i = 1, 2, 3$ such that none of them have a pairwise correlation of $+1$ or $−1$. Can we find a combination of $X_1$ and $X_2$ that has a correlation of $+1$ with $X_3$?

• This is going to depend on the nature of the underlying relationships amongst these variables. BTW, is this homework? If so, please edit to add the homework tag. Oct 21 '12 at 21:23

Yes, it's possible, but not likely to occur in real life. An artificial example

x1 <- rnorm(100)
x2 <- rnorm(100)
x3 <- x1+x2
cor(x1, x2)
cor(x1, x3)
cor(x2, x3)

and, of course, since x3 = x1 + x2, the correlation between (x1 + x2) and x3 is 1.0.

If you had specified that the three variables are independent, I don't think it's possible, but I am not 100% sure.

• If $X_1$ and $X_2$ are uncorrelated with $X_3$, the linear combination $aX_1+bX_2$ is uncorrelated with $X_3$ since covariance is bilinear. Oct 21 '12 at 11:02
• The question didn't say $X_1$ and $X_2$ were uncorrelated with $X_3$, just that the correlations weren't 1.0 or -1.0. Oct 21 '12 at 11:13
• Yes, I was referring to your last line, which talks about what might happen if the variables are pairwise independent, hence uncorrelated. Oct 21 '12 at 11:26
• Oh, sorry. Cool, thanks for that. That was my intuition, but it's nice to know I was right. :-). Oct 21 '12 at 11:30