STA213lecture19.notes

STA213lecture19.notes - Today’s outline: pp 226-231 Order...

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Unformatted text preview: Today’s outline: pp 226-231 Order statistics 1. Definition 5.4.1: The order statistics of a random sample X1 , ..., Xn 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, b the upper integer, and b or [b] the lower integer. Theorem 5.4.3 Let X1 , ..., Xn random sample from a discrete distribution with pmf fX (xi ) = pi , x1 < x2 < ... possible values. Define P0 P1 = P (X ≤ x1 ) P2 = P (X ≤ x2 ) Pi = P (X ≤ xi ) Then the order statistics X(i) are: n = = = ... = 0 p1 p1 + p2 p1 + p2 + ... + pi P (X(j ) ≤ xi ) = k=j n P k (1 − Pi )n−k . ki Theorem 5.4.4 Let X(1) ...X(n) denote order statistics from a continuous distribution with cdf FX (x) and pdf fX (x). Then the pdf of X(j ) is n! n−j fX (x)FX (x)j −1 [1 − FX (x)] . fX(j) (x) = (j − 1)!(n − j )! Example 5.4.5 Uniform order statistics fX(j) (x) = X(j ) ∼ Beta(j, n − j + 1) Theorem 5.4.6 The JOINT distribution of ith and j th order statistics is fX(i) ,X(j) (u, v ) = n! fX (u)fX (v )FX (u)i−1 (i − 1)!(j − 1 − i)!(n − j )! × [FX (v ) − FX (u)] Example 5.4.7: midrange (skipped) j −1−i n! xj −1 (1 − x)n−j , (j − 1)!(n − j )! [1 − FX (v )] n−j 1 Convergence concepts in decreasing strength • Almost sure convergence for X1 , X2 , ... if ∀ P ( lim |Xn − X | < ) n→∞ • Convergence in probability if ∀ n→∞ lim P (|Xn − X | < ) • Convergence in distribution n→∞ lim FXn (x) = FX (x) Weak Law of Large Numbers n→∞ ¯ lim P (|Xn − µ| < ) = 1 ¯ in other words Xn converges in probability to µ. Strong Law of Large Numbers ¯ P ( lim |Xn − µ| < ) = 1 n→∞ ¯ in other words Xn converges almost surely to µ. 2 ...
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This note was uploaded on 01/16/2011 for the course STAT 213 taught by Professor Ioannam during the Fall '09 term at Duke.

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STA213lecture19.notes - Today’s outline: pp 226-231 Order...

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