How to show that normal random variables U1 and U2 are independent?

Question

Prompt:
Assume that $Y_1, Y_2, Y_3$ and $Y_4$ are independently and identically distributed $N(\mu,\sigma^2)$ random variables. Show that $Y_1 + Y_2 – Y_3 – Y_4$ and $Y_1 – Y_2 + Y_3 – Y_4$ are independent.

My first attempt to this question was to let $U_1=Y_1 + Y_2 – Y_3 – Y_4\sim N(0, 4\sigma^2)$ and $U_2= Y_1 – Y_2 + Y_3 – Y_4\sim N(0, 4\sigma^2)$ and try to prove that $$\mathrm{Cov}(U_1,U_2)=\mathrm E(U_1U_2)-\mathrm E(U_1)E(U_2)=0$$

However, I have no idea on how to find out what $E(U_1U_2)$ is. Am I on the right track? Is there any other way to prove that $U_1$ and $U_2$ are independent?

Answer 1

$$E[U_1U_2] = E[(Y_1+Y_2-Y_3-Y_4)(Y_1-Y_2+Y_3-Y_4)]$$ $$=E[Y_1^2]-E[Y_2^2]-E[Y_3^2]+E[Y_4^2]-2E[Y_1Y_4]+2E[Y_2Y3]$$ and the last two terms have the same magnitude so cancel, while the first four terms also cancel

Answer 2

Let $Z_1 = Y_1 - Y_4$ and $Z_2 = Y_2 - Y_3$, then $Z_1$ and $Z_2$ are independent and identically distributed as $N(0,2\sigma^2)$. Now, we can write the two random variables as $$U_1 = Z_1 + Z_2,\quad U_2 = Z_1 - Z_2$$ Then, $$Cov(U_1, U_2) = Cov((Z_1 + Z_2)(Z_1 - Z_2)) = E[(Z_1 + Z_2)(Z_1 - Z_2)] = E[Z_1^2] - E[Z_2^2] = 0$$ The independence of $U_1$ and $U_2$ follows from a well-know fact that two jointly normally distributed random variables are independent if and only if they are uncorrelated.