Suppose we have two independent normal random variables $X$ and $Y$ with mean $1$ and variance $2$. Now consider we have following:
- $X + Y$
- $X - Y$
The question is if $X + Y$ and $X - Y$ are correlated/uncorrelated and independent/not independent (or none of the statements is correct).
Edit: Found suitable formula to check the correlation beetwen given random variables.
$$cov(X+Y)=E((X+Y)(X-Y))+E(X+Y)E(X-Y)$$
$$E(X+Y)=E(X)+E(Y)=2$$ $$E(X-Y)=E(X)-E(Y)=0$$
$$cov(X+Y)=E((X+Y)(X-Y))+2\cdot0=E(X^2-Y^2)$$ $$cov(X+Y)=E(X)E(X)-E(Y)E(Y)=1-1=0$$
If there are no mistakes made, it proves the fact that $X+Y$ and $X-Y$ are not correlated. However, still no thoughts on independence case.
Hint: Being independent, $\ X, Y\ $ are bivariate normally distributed, and hence so are $\ X+Y, X-Y\ $. You can tell whether $\ X+Y\ $ and $\ X-Y\ $ are independent or not by looking at their joint density function, which depends only on their means, variances and correlation coefficient.