Given random sample $X_1, X_2, ..., X_n$ with the distribution $$f(x|\theta) = \left \{ \begin{aligned} e^{-(x-\theta)}, \ \ 0 < x < \theta \\ 0, \text{ otherwise.} \end{aligned} \right. $$
where $\theta \in (-\infty, \infty).$ Show that the estimator $\theta_1 = \min\left\{X_1, X_2, ..., X_n \right\} $ is unbiased.
So, I don't even know where to start, should I randomly generate a minimum for a large amount of normally distributed data and then compare it to the minimum of a sample consisting some arbitrary number of variables? I need to find a solution for the problem in R and I cannot even imagine how to do this.