Let X 1 ,X 2 , . . .,X n be a random sample from a distribution with pdf f(x; ) = x 1 ,

Let X₁,X₂, . . .,X_n be a random sample from a distribution with pdf f(x; θ) = θx^θ−1, 0 < x < 1, zero elsewhere, where θ > 0. Show the likelihood has mlr in the statistic Πⁿ_i=1 X_i. Use this to determine the UMP test for H₀ : θ = θ' against H₁ : θ < θ', for fixed θ' > 0.