Are there non-constant independent random variables $X$ and $Y$ such that $X^2 + Y^2 \equiv 1$?

Question

Are there any independent random variables $X$ and $Y$ such that $X^2 + Y^2 \equiv 1$ and both $X$ and $Y$ are not constants with probability 1?

I tried parametrized functions such as $X = \sin{\theta}, Y = \cos{\theta}$, however it didn't really help, I'm not if there is any example. And if there exists one it must be tricky.

I would appreciate any help.

Answer 1

Yes: suppose you have any $k$, $p$, $q$ all strictly between $0$ and $1$. Having them all $\frac12$ gives some symmetry but is not necessary. Then suppose

• $X = +\sqrt k$ with probability $p$ and $X=-\sqrt k$ with probability $1-p$
• $Y = +\sqrt{1-k}$ with probability $q$ and $Y = -\sqrt{1-k}$ with probability $1-q$, independently of $X$

Then $X$ and $Y$ are independent, neither is almost surely constant, and $X^2+Y^2=1$ with probability $1$