1. Suppose X1, ..., Xn be a random sample from Exp(1) and Y1 < ... <...

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1. Suppose X1, ..., Xn be a random sample from Exp(1) and Y1 < ... <...

1. Suppose X₁, ..., X_n be a random sample from Exp(1) and Y₁ < ... < Y_n be the order statistics from this sample.

a) Find the joint pdf of (Y₁, .. , Y_n).

b) Find the joint pdf of (W₁, .. , W_n) where W₁ = nY₁, W₂ = (n-1)(Y₂ -Y₁), W₃ = (n - 2)(Y₃ - Y₂),..., W_n-1 = 2(Y_n-1 - Y_n-2), W_n = Y_n - Y_n-1.

(c) Show that Wi's are independent and its distribution is identically Exp(1).
(d) Show that nY₁ and sum(X_i-Y₁) are independent.
(e) Show that sum(X_i-Y₁) ~ Gamma(n - 1, 1).
(f) Using (b) and (c), compute E(Y₃).

math Statistics-And-Probability

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