TLFeBOOK FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS T. T. Soong State University of New York at Buffalo, Buffalo, New York, USA TLFeBOOK TLFeBOOK FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS TLFeBOOK TLFeBOOK FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS T. T. Soong State University of New York at Buffalo, Buffalo, New York, USA TLFeBOOK Copyright ! 2004 John Wiley & Sons Ltd, The Atrium, Southern G ate, Chichester, West Sussex PO19 8SQ, England Telephone ( 44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. 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TLFeBOOK To the memory of my parents TLFeBOOK TLFeBOOK Contents PREFACE xiii 1 INTRODUCTION 1 1.1 Organization of Text 2 1.2 Probability Tables and Computer Software 3 1.3 Prerequisites 3 PART A: PROBABILITY AND RANDOM VARIABLES 5 2 BASIC PROBABILITY CONCEPT S 7 2.1 Elements of Set Theory 8 2.1.1 Set Operations 9 2.2 Sample Space and Probability Measure 12 2.2.1 Axioms of Probability 13 2.2.2 Assignment of Probability 16 2.3 Statistical Independence 17 2.4 Conditional Probability 20 Reference 28 Further Reading 28 Problems 28 3 RANDOM VARIABLES A ND PROBABILITY DISTRIBUTIONS 37 3.1 Random Variables 37 3.2 Probability D istributions 39 3.2.1 Probability D istribution Function 39 3.2.2 Probability M ass F unction for D iscrete Random Variables 41 TLFeBOOK [...]... reliance on the use of probabilistic formulations in most scientific disciplines in the future Fundamentals of Probability and Statistics for Engineers T. T Soong ! 2004 John Wiley & Sons, Ltd ISBNs: 0-4 7 0-8 681 3-9 (HB) 0-4 7 0-8 681 4-7 (PB) TLFeBOOK 8 2.1 F undamentals of Probability and Statistics for Engineers ELEMENTS OF SET THEORY Our interest in the study of a random phenomenon is in the statements we can... large For example, the outcome of flipping a coin is not predictable but there is statistical regularity in that the ratio m/n approaches 1 for either 2 Fundamentals of Probability and Statistics for Engineers T. T Soong Ó 2004 John Wiley & Sons, Ltd ISBNs: 0-4 7 0-8 681 3-9 (HB) 0-4 7 0-8 681 4-7 (PB) TLFeBOOK 2 Fundamentals of Probability and Statistics for Engineers heads or tails Random phenomena in scientific... prepared to do this strongly depends on his or her mastery of the fundamentals It is important that the student develop an early appreciation for applications D emonstrations of the utility of this material in nonsuperficial applications not only sustain student interest but also provide the student with stimulation to delve more deeply into the fundamentals Most of the students in engineering and applied... probability and we need the additional information, P (AB) , which is the probability of having between 40 and 50 cars turning left With the statement of three axioms of probability, we have completed the mathematical description of a random experiment It consists of three fundamental constituents: a sample space S , a collection of events A, B, , and the probability function P These three quantities... leads to the recognition that a great deal of uncertainty and variability are inevitably present in problem formulation, and one of the mathematical tools that is effective in dealing with them is probability and statistics Probabilistic ideas are used in a wide variety of scientific investigations involving randomness Randomness is an empirical phenomenon characterized by the property that the quantities... concerning the events that can occur Events and combinations of events thus play a central role in probability theory The mathematics of events is closely tied to the theory of sets, and we give in this section some of its basic concepts and algebraic operations A set is a collection of objects possessing some common properties These objects are called elements of the set and they can be of any kind with any... devote one semester or two quarters to a course of this nature in their programs R ecognizing that the coverage is time limited, it is important that the material be self-contained, representing a reasonably complete and applicable body of knowledge The choice of the contents for this book is in line with the foregoing observations The major objective is to give a careful presentation of the fundamentals. .. Distribution 7.4.2 Chi-Squared Distribution 7.5 Beta and R elated Distributions 7.5.1 Probability Tabulations 7.5.2 Generalized Beta Distribution 7.6 Extreme-Value Distributions 7.6.1 Type-I Asymptotic D istributions of Extreme Values 7.6.2 Type-II Asymptotic Distributions of Extreme Values 7.6.3 Type-III Asymptotic Distributions of Extreme Values 7.7 Summary R eferences F urther R eading and Comments Problems... relative frequency clearly possesses the properties required of the probability measure and is a natural candidate for the probability of A This interpretation is used, for example, in saying that the TLFeBOOK 17 Basic Probability Concepts probability of ‘heads’ in flipping a coin is 1/2 The relative frequency approach to probability assignment is objective and consistent with the axioms stated in Section... that are subsets of space S The union or sum of A and B, denoted by A ‘ B, is the set of all elements belonging to A or B or both S A Figure 2.2 A A and A TLFeBOOK 10 F undamentals of Probability and Statistics for Engineers B A B A (a) A ∪ B (b) A ∩ B Figure 2.3 (a) U nion and (b) intersection of sets A and B The intersection or product of A and B, written as A ’ B, or simply AB, is the set of . TLFeBOOK FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS T. T. Soong State University of New York at Buffalo, Buffalo, New York, USA TLFeBOOK TLFeBOOK FUNDAMENTALS OF PROBABILITY AND STATISTICS. OF PROBABILITY AND STATISTICS FOR ENGINEERS TLFeBOOK TLFeBOOK FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS T. T. Soong State University of New York at Buffalo, Buffalo, New York, USA TLFeBOOK Copyright. important that the student develop an early appreciation for applica- tions. D emonstrations of the utility of this material in nonsuperficial applica- tions not only sustain student interest but