# What is the correlation of $X_1 + X_2$ and $X_1 - X_2$?

Suppose $$E[X_1] = 2$$ and $$Var(X_1) = 4$$. Suppose $$E[X_2] = 0$$ and $$Var(X_2) = 1$$. Suppose also that $$Cor(X_1,X_2) = \frac13$$. I can calculate that the expected value and variance of $$X_1+X_2$$ is $$2$$ and that the variance of $$X_1+X_2$$ is $$6 \frac13$$ and that the expected value and variance of $$X_1-X_2$$ are $$2$$ and $$3 \frac23$$ respectively using $$E[(a_1X_1 + a_2X_2)] = a_1E[X_1] + a_2E[X_2]$$ and $$Var (a_1X_1 + a_2X_2) = a_1^2Var(X_1) + a_2^2Var(X_2) + 2a_1a_2Cov(X_1,X_2)$$

But how do we get the correlation of $$X_1+X_2$$ and $$X_1-X_2$$ ? the solution should be $$0.623$$, but I have no idea how to get there?

$$Cor(X_1+X_2, X_1-X_2) = \frac{Cov(X_1+X_2, X_1-X_2)}{\sqrt{Var(X_1+X_2)Var(X_1-X_2)}}\tag{1}$$
\begin{align}Cov(X_1+X_2, X_1-X_2)&=Cov(X_1,X_1)-Cov(X_1,X_2)\\&+Cov(X_2,X_1)-Cov(X_2,X_2)\\ &= Var(X_1)-Var(X_2)\end{align}
Also, compute $$Var(X_1\pm X_2)$$ and then substitute inside equation $$(1)$$ to evaluate the correlation.