Suppose $X_1 ,X_2 \text{ and } X_3$ are iid normal random variables . Then

Question

Suppose $X_1 ,X_2 \text{ and } X_3$ are iid normal random variables . Then

1. Are $X_1 + X_2 \text{ and } X_1 - X_2 $ independent?

2. Are $X_1 + X_2 + X_3 \text{ and } X_1+X_2 $ independent?

I know that $f_{X_1,X_2}(x_1,x_2) = f_{X_1}(x_1) f_{X_2}(x_2) $ is the condition required for independence . However I am asking for an intuitive answer , which doesnt involve long calculations.

Answer 1

An invertible linear transformation takes jointly normal random variables to jointly normal random variables. Therefore linear combinations of $X_1, X_2, X_3$ are independent if and only if their covariance is $0$.

Answer 2

Robert Israel's observation allows us to answer your question. Explicitly $\operatorname{Cov}(X_i,\,X_j)=\sigma^2\delta_{ij}$ for some $\sigma>0$, so $\operatorname{Cov(}\sum_i a_i X_i,\,\sum_j b_j X_j)=\sigma^2\mathbf{a}\cdot\mathbf{b}$. That means the first example is one of independence but the second isn't.