# Show That $X_1+X_2$ And $X_1-X_2$ Are Independent If $X_1$ And $X_2$ Be Standard Normal Distribution

$$X_1$$ and $$X_2$$ , $$...$$ be variables that have standard normal distribution , How Can We prove the $$X_1+X_2$$ And $$X_1-X_2$$ Are Independent ?

• Are $X_1$ and $X_2$ jointly normal? Maybe independent as well? Jan 13, 2021 at 11:58
• Yes They are independent and normal. Jan 13, 2021 at 12:07
• stats.stackexchange.com/questions/431329 and there's probably more Jan 14, 2021 at 20:42

Since $$A=X_1+X_2, B=X_1-X_2$$ are linear transformations of jointly normal RVs, you'll just need to show $$\operatorname{cov}(X_1+X_2,X_1-X_2)=0$$ in order to show their independence.
Consider that if two random variables $$X$$ and $$Y$$ are independent then $$E[XY]=E[X]E[Y]$$. Now, if $$X$$ and $$Y$$ are jointly normal also the inverse is true.
If $$X_1$$ and $$X_2$$ are jointly normal so is their sum and difference. Therefore, check that $$E[(X_1+X_2)(X_1-X_2)]-E[X_1+X_2]E[X_1-X_2]$$ = 0 holds.
Since$$\left(\begin{matrix}X_1\\X_2\end{matrix}\right)\sim\mathcal N_2\left(\left(\begin{matrix}0\\0\end{matrix}\right),\mathbf I_2\right)$$ where $$\mathbf I_2$$ denote the identity matrix \begin{align*}\left(\begin{matrix}X_1+X_2\\X_1-X_2\end{matrix}\right)&= \left(\begin{matrix}1 &1\\1 &-1\end{matrix}\right)\left(\begin{matrix}X_1\\X_2\end{matrix}\right)\\ &\sim\mathcal N_2\left(\left(\begin{matrix}1 &1\\1 &-1\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right), \left(\begin{matrix}1 &1\\1 &-1\end{matrix}\right)\mathbf I_2 \left(\begin{matrix}1 &1\\1 &-1\end{matrix}\right)\right) \end{align*} and $$\left(\begin{matrix}1 &1\\1 &-1\end{matrix}\right)\mathbf I_2 \left(\begin{matrix}1 &1\\1 &-1\end{matrix}\right)= \left(\begin{matrix}2 &0\\0 &2\end{matrix}\right)$$ shows the absence of correlation.