Solution
Correlation of Sum and Difference
Show the problem again
Random variables $X$ and $Y$ have $\operatorname{Var}(X) = 20$, $\operatorname{Var}(Y) = 5$ and $\operatorname{Cov}(X, Y) = 7/2$. What is the correlation between $X+Y$ and $X-Y$? Give a reduced fraction.
Worked solution
$\operatorname{Cov}(X+Y, X-Y) = \operatorname{Var}X - \operatorname{Var}Y = 15$; the cross terms cancel. Also $\operatorname{Var}(X+Y) = 20 + 5 + 7 = 32$ and $\operatorname{Var}(X-Y) = 25 - 7 = 18$. Their product is $576 = 24^2$, so the correlation is $15/24 = 5/8$.
Source: Original exercise written for AxiomIQ.