Let { f(x;θ): θ>0} denote a family of probability density functions given by f(x;θ) = Γ(2θ)/(Γ(θ))^2 x^(θ-1) ?(1-x)?^(θ-1), 0a) If a random sample of size 1 is observed, find a UMP(Uniformly Most Powerful) Test of size 0.10 for H0: θ=1 vs Ha: θ>1.
Would you reject Ho if x=0.44 is observed?
b) If a random sample of size 2 is observed, demonstrate that a UMP test of size 0.10 for the above test exists. Also decide the form of the rejection region.