# Quiz4Bans - n . Find F Y n ( y ) = P ( Y n ≤ y ), – ∞...

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STAT 410 Spring 2008 Version B Name ANSWERS . Quiz 4 (10 points) Be sure to show all your work, your partial credit might depend on it. No credit will be given without supporting work. 1. Let X 1 , X 2 , … , X n be independent random variables, each with the probability density function f ( x ) = ± ² ³ < - 1 0 1 2 3 x x x . Let Y 1 < Y 2 < … < Y n denote the corresponding order statistics. Let n n n Y Z = . Find the limiting distribution of Z n . a) (2) Find the cumulative distribution function F ( x ) = P ( X 1 x ), – < x < . If x < 1, F ( x ) = P ( X 1 x ) = 0. If x 1, F ( x ) = P ( X 1 x ) = ´ x dy y 1 3 2 = 2 1 1 x - . F ( x ) = P ( X 1 x ) = ± ² ³ < - 1 0 1 1 1 2 x x x

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b) (2) Let Y 1 < Y 2 < … < Y n denote the order statistics based on the random sample X 1 , X 2 , … , X
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Unformatted text preview: n . Find F Y n ( y ) = P ( Y n ≤ y ), – ∞ < y < ∞ . F Y n ( y ) = P ( Y n ≤ y ) = ( F ( y ) ) n = & & ± & & ² ³ < ≥ ´ ´ µ ¶ · · ¸ ¹-1 1 1 1 2 y y y n c) (3) Let n n n Y Z = . Find F n ( z ) = P ( Z n ≤ z ), – ∞ < z < ∞ . F n ( z ) = P ( Z n ≤ z ) = P ( Y n ≤ n z ) = & & ± & & ² ³ < ≥ ´ ´ µ ¶ · · ¸ ¹-n z n z z n n 1 1 1 1 2 d) (3) Find the limiting distribution of Z n . That is, determine the cumulative distribution function F ∞ ( z ) = ∞ → n lim F n ( z ), – ∞ < z < ∞ . ( You probably have not seen this cumulative distribution function before. ) F ∞ ( z ) = ∞ → n lim F n ( z ) = & & ± & & ² ³ ≤ >-2 1 z z e z...
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## This test prep was uploaded on 04/13/2008 for the course STAT 410 taught by Professor Alexeistepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Quiz4Bans - n . Find F Y n ( y ) = P ( Y n ≤ y ), – ∞...

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