Let $X$ and $Y$ be independent random variables. $X$ is a uniform random variable on $[0,1]$. Let $Z=X+Y-\lfloor X+Y\rfloor$. What is the distribution of $Z$?
Let $X_1, X_2, \dots, X_n$ be IID with the same distribution of as $X$, all independent of $Y$. Let $Z_i=X_i+Y-\lfloor X_i+Y\rfloor$ I need to prove that $Z_1, \dots, Z_n$ are IID.
Can anyone point out how to go about solving this problem?