Statistical Inference — range statistic of a shifted uniform family

Let $X_i \sim f_\theta(x) = \begin{cases} 1, & \theta - \frac{1}{2} < x < \theta + \frac{1}{2} \ 0, & \text{otherwise} \end{cases}$ such that $\theta \in (-\infty, \infty)$; $i = 1, 2, …, n$. Consider the following statements: I. The distribution of $X_{(n)} - X_{(1)}$ is independent of $\theta$. II. $E_\theta[(X_{(n)} - X_{(1)}) - E(X_{(n)} - X_{(1)})] = 0$, for all $\theta$. Which of the statements given above is/are correct?