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)


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