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10_06ans - STAT 410 Examples for Summer 2008 In general if...

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STAT 410 Examples for 06/27/2008 Summer 2008 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 ) = ( ) ( ) ( ) ( ) ° = - - ± ± ² ³ ´ ´ µ 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.
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