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- Let X1, · · · , Xn be drawn independently from a distribution with density 1. f(x|θ) = 2/β, θ − 0.25β < x < θ + 0.25β, where β is a positive constant. 2. f(x|θ) = 2x/θ 2 , 0 < x < θ, θ > 0. 3. f(x|θ) = e θ−x for x > θ. For each of the above three cases, construct a 95% confidence interval for θ.Consider a random sample X1,...,Xn,... ∼ iid Beta(θ,1) for n > 2. Prove that the MLE and UMVUE are both consistent estimators for θLet X₁,X₂,...,Xₙ denote a random sample from a distribution that is N(0,θ), where the variance θ is an unknown positive number. Show that there exists a uniformly most powerful test of size α for testing the simple hypothesis H₀ : θ = θ', where θ' is a fixed positive number.