Let $Y_1=\theta+\epsilon_1$ and $Y_2=\theta+\epsilon_2$ where $\epsilon_1$ and $\epsilon_2$ are iid uniform in {-1,1}. Define the confidence set S for $\theta$ as: $S={Y_1-1}$ if $Y_1=Y_2$ and $S={\bar{Y_1}}$ if $Y_1\not=Y_2$. What is the coverage of S?
I've tried this problem multiple times now, So I'm just confusing myself too much at this point (the first time I tried the problem I got 1/2 when I did it over I got 1).
My thoughts are:
coverage($\theta$)=P($\theta\in S)=P(\theta=Y_1-1)P(Y_1=Y_2)+P(\theta=\bar{Y})P(Y_1\not=Y_2) =$ $P(\theta=Y_1-1)P(\epsilon_1=\epsilon_2)+P(\theta=\bar{Y})P(\epsilon_1\not=\epsilon_2)$
This is the point where I start getting lost.
Thanks in advance for any help!