Shifted exponential mean and minimum

Let $\{X_n\}_{n\ge1}$ be a sequence of independent and identically distributed random variables having probability density function $f(x) = \begin{cases} e^{-(x-\theta)} & \text{if } x \ge \theta \ 0 & \text{otherwise,} \end{cases}$ where $\theta > 0$. Consider the following statements: (I) $\frac1n \sum_{i=1}^{n} X_i$ converges in probability to $\frac{\theta+1}{2}$, as $n \to \infty$. (II) $\lim_{n\to\infty} E(\min\{X_1, X_2, …, X_n\}) = \theta$. Which of the above statements is/are true?