Let Y₁ < Y₂ be the order statistics of a random sample of size 2 from a distribution of the continuous type which has pdf f(x) such that f(x) > 0, provided that x ≥ 0, and f(x) = 0 elsewhere. Show that the independence of Z₁ = Y₁ and Z₂ = Y₂ − Y₁ characterizes the gamma pdf f(x), which has parameters α = 1 and β > 0. That is, show that Y₁ and Y₂ are independent if and only if f(x) is the pdf of a Γ(1, β) distribution.