Suppose X1 and X2 are i.i.d. with P{Xi = 1} = P{Xi = +

Suppose X1 and X2 are i.i.d. with P{Xi = θ − 1} = P{Xi = θ + 1} =

1 2 .

Let C be the confidence set consisting of the single point (X1 + X2)/2 if X1 = X2 and X1 − 1 if X1 = X2. Show that, for all θ, Pθ{θ ∈ C} = 0.75 , but Pθ{θ ∈ C|X1 = X2} = 0.5 and Pθ{θ ∈ C|X1 = X2} = 1 .

[Berger and Wolpert (1988)]