# chap1f - Probability and Statistics with Reliability,...

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Copyright © 2003 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 1: Introduction Dept. of Electrical & Computer Engineering Duke University Email: kst@ee.duke.edu URL: www.ee.duke.edu/~kst

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Copyright © 2003 by K.S. Trivedi 2 Need to Model Random Phenomena ± Random Phenomena in a Computer/networking Environment ± Arrival of jobs/messages/requests. ± Execution (transmission/processing) time of jobs/messages/requests. ± Memory requirement of jobs/messages/requests. ± Failure or repair of components or resources. ± How to Quantify Randomness? ± Use probabilistic models. ± How to Estimate these Quantifiers? ± Use statistical techniques.
Copyright © 2003 by K.S. Trivedi 3 Modeling Random Phenomena Measurement Data Statistical Analysis Model Input Parameters Probability Model (PM) Input Output Validation Model Outputs

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Copyright © 2003 by K.S. Trivedi 4 Components of a Probability Model 1. Sample Space (S): A set of all possible observable “states” of a random phenomena. 2. Set of Events ( ): A set of all possible events of interest. 3. Probability of Events (P): A consistent description of the likelihood of observing an event. Thus a PM is a triple: PM = (S, , P).
Copyright © 2003 by K.S. Trivedi 5 Sample Space ± Probability of an event is meant to represent the relative likelihood that an outcome of an experiment will result in occurrence of that event. ± It implies random experiments. ± A random experiment can have many possible outcomes; each outcome is known as a sample point ( a.k.a. elementary event) has some probability assigned. This assignment may be based on measured data or guestimates (“equally likely” is a convenient and often made assumption). ± Sample Space S : a set of all possible outcomes (elementary events) of a random experiment. ± Finite (e.g., if statement execution; two outcomes) ± Countable (e.g., number of times a while statement is executed; Sample space is either finite or countably infinite) ± Continuous (e.g., time to failure of a component)

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Copyright © 2003 by K.S. Trivedi 6 Events ± An event E is a collection of zero or more sample points from S. Event E is a subset of S . ± S is the universal event and the empty set is represented by F F,E E E,F E E S Explained in next slide
Copyright © 2003 by K.S. Trivedi 7 Terminology and Definitions ± S and E are sets hence can use of set operations. ± E or F : : Union of two events ± E and , EF : Intersection ± Not E: Ē : Complement F E F E n i n i E ... E E E 1 2 1 = = n i n n i ...E E E E ... E E E 1 2 1 2 1 = = =

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Copyright © 2003 by K.S. Trivedi 8 Algebra of events ± Sample space is a set and events are the subsets of this (universal) set.
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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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chap1f - Probability and Statistics with Reliability,...

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