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Lecture_22

Lecture_22 - 1 ORDER STATISTICS Lecture 22 ORIE3500/5500...

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Unformatted text preview: 1 ORDER STATISTICS Lecture 22 ORIE3500/5500 Summer2009 Chen Class Today • Order Statistics • Statistics 1 Order Statistics We have met the order statistics of uniformly distributed random variables in our prelim. Here we will learn how to formally treat the order statistics of the generally distributed random variables. Suppose X 1 ,X 2 ,...,X n are i.i.d. random variables with density function f X and cdf F X . The ordered random variables are denoted by X ( i ) ,i = 1 ,...,n which are just the random variables X i ’s ordered such that X (1) ≤ X (2) ≤ ··· ≤ X ( n ) . In particular X (1) = min( X 1 ,...,X n ) and X ( n ) = max( X 1 ,...,X n ). We have already seen before that F X (1) ( x ) = 1- (1- F X ( x )) n . and taking derivative we get f X (1) ( x ) = n (1- F X ( x )) n- 1 f X ( x ) . Example If X 1 ,...,X n are i.i.d N (0 , 1) random variables , then the density of X (1) is given by f X (1) ( x ) = n √ 2 π ‡ 1- Φ( x ) · n- 1 e- x 2 / 2 Similarly the cdf of the n th order statistic is F X ( n ) ( x ) = ( F X ( x )) n , 1 1 ORDER STATISTICS...
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Lecture_22 - 1 ORDER STATISTICS Lecture 22 ORIE3500/5500...

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