Let $X_1,...,X_n$, $n\geq5$ be a random sample from a distribution with the density function:
$f(x,\theta)= e^{-(x-\theta)}$ where $x\geq\theta$. Then, which of the following statements are correct:
A $95%$% confidence interval of $\theta$ has to be of finite length.
[$X_{(1)}+\frac{1}{n}ln(0.05),X_{(1)}]$ is a 95% confidence interval for $\theta$.
A $95%$ confidence interval of $\theta$ can be of length 1.
A $95%$ confidence interval of $\theta$ can be length 2.
Here, $X_{(1)}$ is the first ordered Statistic.
My approach
Clearly, the second option is correct. It can be checked by adjusting the term and integrating the density of first ordered Statistic. But I am not sure, what does the finite length mean. If I proceed from the interval, that we obtained from the second option, then the length of $95%$ Confidence interval is turning out to be $\frac{1}{n}ln(0.05)$. Also, the value of $n$ cannot be less than five. So, clearly the length cannot be 1 or two. Thus, we conclude that the only option that seam reasonable is $2$. Please check my steps and correct me if I am wrong. Thanks