Correlation of sum and difference of linked variables

Question

Random variables $X$ and $Y$ have correlation exactly $1$ , and the standard deviation of $Y$ is smaller than that of $X$ . What is the correlation between $X+Y$ and $X-Y$ ?

Short answer

$1$ . Perfect correlation means $Y=cX+d$ with $c=\sigma_Y/\sigma_X\in(0,1)$ . Then $X+Y=(1+c)X+d$ and $X-Y=(1-c)X-d$ are both increasing linear functions of $X$ , so they are perfectly positively correlated.

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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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