Independent $\displaystyle X, Y$ both have std. normal distribution.

Which statements are correct?

- $\displaystyle P(X>0)=P(Y<0)$
- $\displaystyle X+Y$ is independent of $\displaystyle X-Y$
- $\displaystyle X-Y$ is normally distributed
- $\displaystyle X+Y$ is std. normally distributed
- $\displaystyle P(X>0, Y>0)=\frac{1}{4}$
- $\displaystyle P(X>Y)=P(Y>X)$
- $\displaystyle X+Y$ is independent of $\displaystyle X$
- $\displaystyle X^2+Y^2=1$
T= true,F= false, blank = "I have (really) no idea."

T: $\displaystyle P(X>0)=\frac{1}{2}=P(Y<0)$F: "intuition" ... really,why(if "F" is correct, of course)?T: solved it on MHF in Normal distribution?F: follows from3.?T: $\displaystyle =P(X>0)P(Y>0)=\frac{1}{2}\frac{1}{2}=\frac{1}{4}$T: again, "intuition"F: I guess $\displaystyle X+Y$ is most strongly dependent on $\displaystyle X$?- : no clue (
at all)

Help me with argumentingwhyit isT(orF) ... correct any flawed reasoning (there are many).