Modulo Sum of Random Variables I'm looking for proof using CDFs or MGFs to show for $X$ and $Y$ independent random variables,
$$
X \sim \mathrm{Unif}(0,1) \text{ and } Y \sim \mathrm{Unif}(0,1) \implies (X+Y \bmod 1) \sim \mathrm{Unif}(0,1).
$$
I've tried using MGFs for $X+Y$ but can't seem to figure out how to incorporate the $\bmod 1$ aspect.  I've read this thread but I'm really looking for a proof using MGFs (or CDFs).  Any hints are appreciated!

You want $P(Z\leq z) = P(\{X+Y \leq z\} \cup \{1<X+Y\leq 1+z\})$, this is equal to
$$\int^z_{y=z-x}\int^z_{x=0}1\text{ d}x\text{d}y + \int_{y=1-z}^{1+z-x}\int_{x=z}^11\text{ d}x\text{d}y$$
You need to do the integral with respect to $y$ first, then $x$.
also note: MGF does not work. MGF only works with composition of distributions/sum of distributions. Taking mod is not particularly nice.
