# If $X_i \sim U(\theta-\frac{1}{2};\theta+\frac{1}{2})$, show that $[X_{(1)},X_{(n)}]$ is a confidence interval

Let $$X_1,...X_n$$ random sample from $$f(x;\theta)=I_{[\theta-\frac{1}{2};\theta+\frac{1}{2}]}(x)$$.

a) Show that $$[X_{(1)},X_{(n)}]$$ is a confidence interval for $$\theta$$.

b) Compute the expected length $$E(X_{(n)}-X_{(1)})$$

c) Find the confidence level.

a) I don't know how to show that $$[X_{(1)},X_{(n)}]$$ is confidence interval, I think I should find a pivotal quantity with $$X_{(1)}$$ and $$X_{(n)}$$ and show that its distibution doesn't depend on $$\theta$$. Is it right?

b) I found the density function of $$X_{(n)}-X_{(1)}$$ (Calculations are too large to write here) and get that $$E(X_{(n)}-X_{(1)})= \frac{n-1}{n+1}$$

c)$$P(X_{(1)}\leq \theta \leq X_{(n)})=P(X_{(n)}\geq\theta)-P(X_{(1)}\geq \theta)=(1-F_{X_{(n)}}(\theta))-(1-F_{X_{(1)}}(\theta))$$ and just have to compute this probabilities.

In b) and c) I'm almost sure that it's correct, but I just don't not how to do a)