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 $X_3$?
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$\begingroup$ 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. $\endgroup$– gung - Reinstate MonicaOct 21 '12 at 21:23
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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.
answered Oct 21 '12 at 10:19
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4$\begingroup$ 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. $\endgroup$ Oct 21 '12 at 11:02
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$\begingroup$ 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. $\endgroup$ Oct 21 '12 at 11:13
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$\begingroup$ Yes, I was referring to your last line, which talks about what might happen if the variables are pairwise independent, hence uncorrelated. $\endgroup$ Oct 21 '12 at 11:26
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$\begingroup$ Oh, sorry. Cool, thanks for that. That was my intuition, but it's nice to know I was right. :-). $\endgroup$ Oct 21 '12 at 11:30