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 0.9 − 2θ 0.1 − θ

where 0 < θ < 0.1. For testing H : θ = 0.05 against θ > 0.05 at level α = 0.05, determine which of the following tests (if any) is UMP:

(i) φ(0) = 1, φ(1) = φ(2) = φ(3) = 0;

(ii) φ(1) = 0.5, φ(0) = φ(2) = φ(3) = 0;

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