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 $\displaystyle X, Y,$ and $\displaystyle Z$ are nonnegative random variables such that $\displaystyle P(X+Y+Z \leq 1.3)=1$. Show that $\displaystyle X, Y,$ and $\displaystyle Z$ cannot possibly have a join distribution under which each of their marginal distributions is the uniform distribution on the interval $\displaystyle [0,1]$.

Work:

$\displaystyle \int_0^1\int_0^1 f(x,y,z)dxdy=z$

$\displaystyle \int_0^1\int_0^1 f(x,y,z)dxdz=y$

$\displaystyle \int_0^1\int_0^1 f(x,y,z)dydz=x$

$\displaystyle \int_0^1 xdx=\frac{1}{2}$

$\displaystyle \int_0^1 ydy=\frac{1}{2}$

$\displaystyle \int_0^1 zdz=\frac{1}{2}$

Answer:

Integrating $\displaystyle f(x,y,z)$ over the cube gives $\displaystyle p=\frac{1}{2}$ instead of $\displaystyle 1$, therefore we have a contradiction. So $\displaystyle X, Y,$ and $\displaystyle Z$ cannot possibly have a join distribution under which each of their marginal distributions is the uniform distribution on the interval $\displaystyle [0,1]$.