Let the distribution of X be given by x 01 2 3 P(X = x) 2

Let the distribution of X be given by x 01 2 3 Pθ(X = x) θ 2θ .9 − 2θ .1 − θ

where 0 <θ<.1. For testing H : θ = .05 against θ>.05 at level α = .05, determine which of the following tests (if any) is UMP:

(i) φ(0) = 1, φ(1) = φ(2) = φ(3) = 0;

(ii) φ(1) = .5, φ(0) = φ(2) = φ(3) = 0;

(iii) φ(3) = 1, φ(0) = φ(1) = φ(2) = 0.