Let the random variable X have the pdf f(x; ) = (1/)e x/ , 0 < x < , zero elsewhere. Consider the simple hypothesis

Let the random variable X have the pdf f(x; θ) = (1/θ)e^−x/θ, 0 < x < ∞, zero elsewhere. Consider the simple hypothesis H₀ : θ = θ' = 2 and the alternative hypothesis H₁ : θ = θ'' = 4. Let X₁,X₂ denote a random sample of size 2 from this distribution. Show that the best test of H₀ against H₁ may be carried out by use of the statistic X₁ + X₂.