Let Y₁ < Y₂ < · · · < Y_n be the order statistics of a random sample of size n from the exponential distribution with pdf f(x) = e^−x, 0 < x < ∞, zero elsewhere.

(a) Show that Z₁ = nY₁, Z₂ = (n−1)(Y₂ −Y₁), Z₃ = (n−2)(Y₃ −Y₂), . . . , Z_n = Y_n−Y_n−1 are independent and that each Z_i has the exponential distribution.

(b) Demonstrate that all linear functions of Y₁, Y₂, . . . , Y_n, such as Σⁿ₁ a_iY_i, can be expressed as linear functions of independent random variables.