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I know that the sum of two normally distributed random variables is also normal.

But what about the opposite? Can the sum (or substraction) of two non-normally distributed random variables be normal?

X-Y=Z Can Z be normal if X and Y are not? Any example?

I came to this question while thinking about the need of checking normallity of the differences of paired measures.

The question Is joint normality a necessary condition for the sum of normal random variables to be normal? is not the same than mine, it speaks about joint normallity.

And this one Sum of independent non-normal random variables says it's a duplicate but I can't find the original one.

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    $\begingroup$ Your question is unclear - in what way (if any) are the two random variables related? Can they be dependent? Or only independent? $\endgroup$
    – Glen_b
    Oct 4, 2016 at 9:20

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As you've quoted, joint normality is a sufficient condition for $X+Y$ to be normal. The general case is not true. We can have $X+Y$ to be non-normal with $X$ and $Y$ normal. The simple example is to have $X=-Y$ where $X,Y$ will be normal and $X+Y$ isn't.

Similarly, to make a sum normal out of two non-normal, you just need $X = \frac12A+B$ and $Y = \frac12A-B$ where $A$ is normal and $B$ follows whatever strange distribution.

And the linked question answers the independent $X$ and $Y$ case. Not sure why you can't find the link, but it's here

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    $\begingroup$ $X+Y$ indeed is Normal when $X=-Y$: it has a standard deviation of zero. It would be awful if we had to modify all the theorems that assert linear combinations of jointly Normal variates are Normal just to rule out this special case! $\endgroup$
    – whuber
    Oct 4, 2016 at 18:03

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