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Let {x1, x2, . . . , xn} be a set of real numbers and

Fundamentals of Probability, with Stochastic Processes | 3rd Edition | ISBN: 9780131453401 | Authors: Saeed Ghahramani ISBN: 9780131453401 223

Solution for problem 21 Chapter 11.3

Fundamentals of Probability, with Stochastic Processes | 3rd Edition

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Fundamentals of Probability, with Stochastic Processes | 3rd Edition | ISBN: 9780131453401 | Authors: Saeed Ghahramani

Fundamentals of Probability, with Stochastic Processes | 3rd Edition

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Problem 21

Let {x1, x2, . . . , xn} be a set of real numbers and define x = 1 n .n i=1 xi, s2 = 1 n 1 .n i=1 (xi x) 2 . Prove that at least a fraction 1 1/k2 of the xis are between x ks and x + ks. Sketch of a Proof: Let N be the number of x1, x2, . . . , xn that fall in A = [ x ks, x + ks]. Then s2 = 1 n 1 .n i=1 (xi x) 2 1 n 1 . xi,A (xi x) 2 1 n 1 . xi,A k2 s2 = n N n 1 k2 s2 . This gives (N 1)/(n 1) 1 (1/k2). The result follows since N/n (N 1)/(n 1).

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Assignment # 2 STA 5205, 5126 and 4202 Date: Monday, October 5, 2015 This assignment is based on Chapters 3 and 4. You must show all necessary work to get full credit. Please submit your assignment in due time. Some selected questions will be graded. Pl keep 1” margin in all sides of the paper. You must define both null and alternative hypotheses for any test related question. You may use any computer software unless oherwise stated. First page is your cover page. First & Last Name:----------------------------------Panther ID:------------------------------------------- Problem #1: Four chemists are asked to determine the percentage of methyl alcohol in a certain chemical compound. Each chemist makes three determination

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Textbook: Fundamentals of Probability, with Stochastic Processes
Edition: 3
Author: Saeed Ghahramani
ISBN: 9780131453401

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Let {x1, x2, . . . , xn} be a set of real numbers and