If $\frac{X+Y}{\sqrt{2}}\stackrel{d}{=}X\stackrel{d}{=}Y$, use the CLT to show that $X,Y\in N(0,1)$
Suppose X and Y are i.i.d and $\frac{X+Y}{\sqrt{2}}\overset{d}=X\overset{d}=Y$, show that X has a normal distribution.