If $X + Y$ has the distribution you would expect from independence, then are $X$ and $Y$ independent?

Question

Suppose that $X$ and $Y$ are random variables, and that $X'$ has the same distribution as $X$, and $Y'$ the same distribution of $Y$. Suppose also that $X'$ and $Y'$ are independent. Then if $X + Y$ has the same distribution as $X' + Y'$, are $X$ and $Y$ also independent?

Answer 1

Let the discrete variables $X,Y$ each have range $\{1,2,3,4\}$ with joint probability mass function $P(X=x,\ Y=y)=f(x,y)$ where $$f(1,2)=f(2,4)=f(3,1)=f(4,3)=0,$$ $$f(1,3)=f(2,1)=f(3,4)=f(4,2)=\frac18,$$ $$f(1,1)=f(1,4)=f(2,2)=f(2,3)=f(3,2)=f(3,3)=f(4,1)=f(4,4)=\frac1{16}.$$ You can verify that $X$ and $Y$ have a uniform distribution on $\{1,2,3,4\}$ and that $X+Y$ has the same distribution it would have if $X$ and $Y$ were independent.