# chap4f - Probability and Statistics with Reliability...

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Copyright © 2005 by K.S. Trivedi 1 Probability and Statistics with Reliability, Queuing and Computer Science Applications Second edition by K.S. Trivedi Publisher-John Wiley & Sons Chapter 4 :Expectation Dept. of Electrical & Computer engineering Duke University Email: [email protected] URL: www.ee.duke.edu/~kst

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Copyright © 2005 by K.S. Trivedi 2 Expected (Mean, Average) Value ± There are several ways to abstract information from the CDF into a single number: Median, Mode and mean. ± Mean: (when X is discrete) (when X is continuous) ± Using integration by parts the following formula for E[X] can be obtained so that it can be directly computed using the distribution function: = k k X k x p x X E ) ( ] [ = dx x xf X ) ( = 0 0 ) ( )] ( 1 [ ] [ dx x F dx x F X E X X
Copyright © 2005 by K.S. Trivedi 3 Expected (Mean, Average) Value Note that the expression for E[X] may not always converge; that is E[X] may not always exist in some cases. For example: ± Problem 1 on p. 196 of text ± A Defective random variable (p. 146 of text) ± The Pareto distribution with α ≤ 1 (p. 227 of text)

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Copyright © 2005 by K.S. Trivedi 4 Linear Search Problem ± Consider the problem of searching for a specific name in the table. ± We scan the table sequentially, starting from one end, until we either find the name or reach the other end, indicating that the required name is missing from the table. ± Refer Pg. 194 of Text for code in C. ± Let X be the discrete random variable denoting the number of comparisons “myName = Table[I]” made.
Copyright © 2005 by K.S. Trivedi 5 Linear Search Problem (contd.) ± The set of all possible values of X is {1, 2, . . , n + 1} and X = n + 1 for unsuccessful searches. ± Consider a random variable Y denoting the number of comparisons on a successful search. The set of all possible values of Y is {1, 2, . . , n}. ± Assume pmf of Y to be uniform over the range

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Copyright © 2005 by K.S. Trivedi 6 Linear Search Problem (contd.) ± Thus, on the average, approximately half the table needs to be searched.
Copyright © 2005 by K.S. Trivedi 7 Linear Search Problem (contd.) ± Assumption of discrete uniform distribution for linear search rarely holds ± Assume that table search starts from the front. ± denotes the access probability for name Table[i], then average successful search time is ± E[Y] is minimized when ± Many tables in practice follow Zipf’s law

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Copyright © 2005 by K.S. Trivedi 8 Linear Search Problem (contd.) ± constant c is determined by 1 where, and C is Euler constant ± Average search time is: for large n, E[Y] is considerably lesser than (n+1)/2
Copyright © 2005 by K.S. Trivedi 9 Zipf’s law ± Zipf’s law has been used to model the distribution of Web page requests. ± p Y (i), the probability of a request for the i th most popular page is inversely proportional to i

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Copyright © 2005 by K.S. Trivedi 10 Zipf’s law (contd.) ± Assumption ± Web page requests are independent ± The cache can hold only m Web pages regardless of the size of each Web page.
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## This note was uploaded on 04/08/2010 for the course COMPUTER E 409232 taught by Professor Mohammadabdolahiazgomiph.d during the Spring '10 term at Islamic University.

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chap4f - Probability and Statistics with Reliability...

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