Let Y 1 Y 2 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 Z 1 nY 1 , Z 2 (n1)(Y 2 Y 1 ), Z 3 (n2)(Y 3 Y 2 ), , 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 Y 1 , Y 2 , , Y n , such as n 1 a i Y i , can be expressed as linear functions of independent random variables

Question: Let Y 1 < Y 2 < < Y n be the order statistics of a random sample of size n from

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.

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