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I'm told that $X$ has uniform distribution on $(-1,1)$ and given $X=x$, $Y$ is uniformly distributed on $(-\sqrt{1-x^2},\sqrt{1-x^2})$. I know it's not uniformly distributed over the unit disk $\{(x,y):x^2+y^2\lt 1\}$, because my professor did this in class but I didn't quite understand what he was talking about. For a reason why, it was because I was just a few minutes late for class and he doesn't like to explain things twice. So could anyone show how he did this?

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up vote 1 down vote accepted

Over the unit disc, a uniform distribution should have the same probability for any portion of that disc with the same area, hence the name uniform. That's not the case for the example your professor gave you.

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If you draw (X,Y) uniformly from the unit circle, would the marginal of X be uniform?

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