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STA213lecture19.notes

# STA213lecture19.notes - Todays outline pp 226-231 Order...

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Today’s outline: pp 226-231 Order statistics 1. Definition 5.4.1: The order statistics of a random sample X 1 , ..., X n are the sample values placed in ascending order, denoted X (1) , ..., X ( n ) . 2. Range, Median, upper/lower quartile, interquartile range. 3. Definition 5.4.2: We define { b } , nearest integer, d b e the upper integer, and b b c or [ b ] the lower integer. Theorem 5.4.3 Let X 1 , ..., X n random sample from a discrete distribution with pmf f X ( x i ) = p i , x 1 < x 2 < ... possible values. Define P 0 = 0 P 1 = P ( X x 1 ) = p 1 P 2 = P ( X x 2 ) = p 1 + p 2 ... P i = P ( X x i ) = p 1 + p 2 + ... + p i Then the order statistics X ( i ) are: P ( X ( j ) x i ) = n X k = j n k P k i (1 - P i ) n - k . Theorem 5.4.4 Let X (1) ...X ( n ) denote order statistics from a continuous distribution with cdf F X ( x ) and pdf f X ( x ). Then the pdf of X ( j ) is f X ( j ) ( x ) = n ! ( j - 1)!( n - j )! f X ( x ) F X ( x ) j - 1 [1 - F X ( x )] n - j . Example 5.4.5 Uniform order statistics f X ( j ) ( x ) = n ! ( j - 1)!( n - j )! x j - 1 (1 - x ) n - j , X ( j ) Beta ( j, n - j + 1) Theorem 5.4.6 The JOINT distribution of i th and j th order statistics is

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STA213lecture19.notes - Todays outline pp 226-231 Order...

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