I'm not sure if I proved the following correctly. (It is in a chapter on expectation, and I didn't use anything about expectation.) Any feedback would be greatly appreciated.

Problem:

Suppose that and are nonnegative random variables such that . Show that and cannot possibly have a join distribution under which each of their marginal distributions is the uniform distribution on the interval .

Work:

Answer:

Integrating over the cube gives instead of , therefore we have a contradiction. So and cannot possibly have a join distribution under which each of their marginal distributions is the uniform distribution on the interval .