Correlation of sum and difference under rho of -1

Question

Random variables $X$ and $Y$ have correlation $\rho(X,Y)=-1$ , and the standard deviation of $Y$ is strictly greater than that of $X$ , i.e. $\sigma_Y>\sigma_X$ . What is the correlation between $X+Y$ and $X-Y$ ?

Short answer

$-1$ . Perfect negative correlation means $Y=aX+b$ with $a<0$ , and $\sigma_Y>\sigma_X$ forces $a<-1$ . Then $X+Y=(1+a)X+b$ has a negative coefficient on $X$ while $X-Y=(1-a)X-b$ has a positive one, so the two are exact linear functions of $X$ moving in opposite directions.

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Lessons behind this question

Covariance and Correlation
Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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