Let $U \sim unif[0,1]$ and $U_n = \frac{\lfloor nU\rfloor}{n}$.
a) Determine the distribution of the difference variable $W_n = U - U_n$.
b) Using part a), evaluate the correlation coefficient $\rho(U,U_n)$.
c) For $Y=U$ and $X=U_n$, obtain the unique $\alpha,\beta\in\mathbb{R}$ and error variable $Z$ such that $Y = \alpha + \beta X + Z$ with $E(Z)=0$ and $\rho(X,Z)=0$.
