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

Question

$X_1$ 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 ?

Answer 1

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.

Answer 2

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.

Answer 3

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.