Let Y 1 < Y 2 be the order statistics of a random sample of size 2

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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.

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Introduction To Mathematical Statistics

ISBN: 9780321794710

7th Edition

Authors: Robert V., Joseph W. McKean, Allen T. Craig

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