EX 20 - STAT 400 Spring 2011 Examples for 05/02/2011 In...

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STAT 400 Examples for 05/02/2011 Spring 2011 In general, if X 1 , X 2 , … , X n is a random sample of size n from a continuous distribution with cumulative distribution function F ( x ) and probability density function f ( x ), then F max X i ( x ) = P ( max X i x ) = P ( X 1 x , X 2 x , … , X n x ) = P ( X 1 x ) P ( X 2 x ) P ( X n x ) = ( ) ( ) n x F . f max X i ( x ) = F ' max X i ( x ) = ( ) ( ) ( ) 1 F x f x n n - . 1 – F min X i ( x ) = P ( min X i > x ) = P ( X 1 > x , X 2 > x , … , X n > x ) = P ( X 1 > x ) P ( X 2 > x ) P ( X n > x ) = ( ) ( ) n x F 1 - . F min X i ( x ) = ( ) ( ) n x F 1 1 - - . f min X i ( x ) = F ' min X i ( x ) = ( ) ( ) ( ) 1 F 1 x f x n n - - . Let Y k = k th smallest of X 1 , X 2 , … , X n . F Y k ( x ) = P ( Y k x ) = P ( k th smallest observation x ) = P ( at least k observations are x ) = ( ) ( ) ( ) ( ) = - - n k i i n i x x i n F 1 F . f Y k ( x ) = F ' Y k ( x ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) F 1 F 1 1 ! ! ! x f x x k n k n k n k - - - - - .
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1. Let X 1 , X 2 , X 3 , X 4 be a random sample ( i.i.d. ) of size n = 4 from a probability distribution with the p.d.f. f ( x ) = 3 / x 4 , x > 1. Let Y k = k th smallest of X 1 , X 2 , … , X n . For x 1, F ( x ) = 0. For x > 1, F ( x ) = 3 1 3 1 4 1 1 1 3 x y dy y x x - = - = . a) Find P ( Y 4 < 1.75 ) = P ( max X i < 1.75 ). P ( max X i < 1.75 ) = P ( X 1 < 1.75, X 2 < 1.75, X 3 < 1.75, X 4 < 1.75 ) = F ( 1.75 ) 4 = ( 1 – 1 / 1.75 3 ) 4 0.4377643 . b) Find P ( Y 4 > 2 ) = P ( max X i > 2 ). P ( max X i > 2 ) = 1 – P ( max X i 2 ) = 1 – P ( X 1 2, X 2 2, X 3 2, X 4 2 ) = 1 – F ( 2 ) 4 = 1 – ( 1 – 1 / 2 3 ) 4 = 1 – ( 7 / 8 ) 4 0.41381836 .
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This note was uploaded on 11/28/2011 for the course STAT 400 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.

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EX 20 - STAT 400 Spring 2011 Examples for 05/02/2011 In...

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