I don't understand your integrals.
IF you integrate out the other variables you obtain the marginal density.
Hence
IF you think that Z is U(0,1)
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 .