Question

Let $X_1$ and $X_2$ i.i.d. Uniformly distributed over [$ \theta -\frac {1} {2},\theta +\frac {1} {2}] $ for an unknown $\theta$ in $\mathbb{R}$. Show that [min($ X_1,X_2$), max($ X_1,X_2$)]  is 50% CI for $\theta$.

hint: u don’t have to calculate the distribution of $\min \left( X_{1},X_{2}\right) $ or $\max \left( X_{1},X_{2}\right) $

Answer 1

X1 ~ U (theta - 1/2, they +1/2)

X2~ U( theta - 1/2, theta +1/2)

We have to show that [min( X1, X2), max(X1, X2)] is 50% CI for theta

Let, theta = 1

Then,

CI = [1-1/2,1+1/2]=[1/2,3/2]

So here if you see lower limit it is 1/2 so, theta - 1/2=1-1/2=1/2 which is differ by 1/2 = 50%

It can be generalised to other values of theta also.

Hence, [min(X1, X2) ,max(X1,X2)] is 50 % CI for theta.