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A First Course in Probability

ISBN: 9780137463145 | 0137463146
Format: Hardcover
Publisher: PRENTICE HALL
Pub. Date: 8/1/1997

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SummaryTable of Contents
This market leader is written as an elementary introduction to the mathematical theory of probability for students in mathematics, engineering, and the sciences who possess the prerequisite knowledge of elementary calculus. A major thrust of the Fifth Edition has been to make the book more accessible to today's students. The exercise sets have been revised to include more simple, mechanical problems and a new section of Self-Test Problems with fully worked out solutions conclude each chapter. In addition, many new applications have been added to demonstrate the importance of probability in real situations. A software diskette, referenced in text and packaged with each copy of the book, provides an easy to use tool for students to derive probabilities for binomial, Poisson, and normal random variables, illustrate and explore the central limit theorem, work with the strong law of large numbers, and more.
PREFACExi
1 COMBINATORIAL ANALYSIS
1(24)
1.1 Introduction
1(1)
... MORE1.2 The Basic Principle of Counting
2(1)
1.3 Permutations
3(2)
1.4 Combinations
5(5)
1.5 Multinomial Coefficients
10(2)
1.6 On the Distribution of Balls in Urns*
12(3)
Summary
15(1)
Problems
16(3)
Theoretical Exercises
19(4)
Self-Test Problems and Exercises
23(2)
2 AXIOMS OF PROBABILITY
25(42)
2.1 Introduction
25(1)
2.2 Sample Space and Events
25(5)
2.3 Axioms of Probability
30(2)
2.4 Some Simple Propositions
32(4)
2.5 Sample Spaces Having Equally Likely Outcomes
36(12)
2.6 Probability As a Continuous Set Function*
48(4)
2.7 Probability As a Measure of Belief
52(1)
Summary
53(1)
Problems
54(7)
Theoretical Exercises
61(3)
Self-Test Problems and Exercises
64(3)
3 CONDITIONAL PROBABILITY AND INDEPENDENCE
67(59)
3.1 Introduction
67(1)
3.2 Conditional Probabilities
67(5)
3.3 Bayes' Formula
72(11)
3.4 Independent Events
83(13)
3.5 P(.\F) is a Probability(*)
96(7)
Summary
103(1)
Problems
104(14)
Theoretical Exercises
118(5)
Self-Test Problems and Exercises
123(3)
4 RANDOM VARIABLES
126(66)
4.1 Random Variables
126(5)
4.2 Distribution Functions
131(3)
4.3 Discrete Random Variables
134(2)
4.4 Expected Value
136(3)
4.5 Expectation of a Function of a Random Variable
139(3)
4.6 Variance
142(2)
4.7 The Bernoulli and Binomial Random Variables
144(10)
4.7.1 Properties of Binomial Random Variables
149(3)
4.7.2 Computing the Binomial Distribution Function
152(2)
4.8 The Poisson Random Variable
154(8)
4.8.1 Computing the Poisson Distribution Function
161(1)
4.9 Other Discrete Probability Distribution
162(9)
4.9.1 The Geometric Random Variable
162(2)
4.9.2 The Negative Binomial Random Variable
164(3)
4.9.3 The Hypergeometric Random Variable
167(3)
4.9.4 The Zeta (or Zipf) distribution
170(1)
Summary
171(2)
Problems
173(11)
Theoretical Exercises
184(5)
Self-Test Problems and Exercises
189(3)
5 CONTINUOUS RANDOM VARIABLES
192(52)
5.1 Introduction
192(3)
5.2 Expectation and Variance of Continuous Random Variables
195(5)
5.3 The Uniform Random Variable
200(4)
5.4 Normal Random Variables
204(11)
5.4.1 The Normal Approximation to the Binomial Distribution
212(3)
5.5 Exponential Random Variables
215(7)
5.5.1 Hazard Rate Functions
220(2)
5.6 Other Continuous Distributions
222(5)
5.6.1 The Gamma Distribution
222(2)
5.6.2 The Weibull Distribution
224(1)
5.6.3 The Cauchy Distribution
225(1)
5.6.4 The Beta Distribution
226(1)
5.7 The Distribution of a Function of a Random Variable
227(3)
Summary
230(2)
Problems
232(5)
Theoretical Exercises
237(4)
Self-Test Problems and Exercises
241(3)
6 JOINTLY DISTRIBUTED RANDOM VARIABLES
244(65)
6.1 Joint Distribution Functions
244(8)
6.2 Independent Random Variables
252(2)
6.3 Sums of Independent Random Variables
264(8)
6.4 Conditional Distributions: Discrete Case
272(1)
6.5 Conditional Distributions: Continuous Case
273(3)
6.6 Order Statistics*
276(4)
6.7 Joint Probability Distribution of Functions of Random Variables
280(8)
6.8 Exchangeable Random Variables*
288(3)
Summary
291(2)
Problems
293(7)
Theoretical Exercises
300(5)
Self-Test Problem and Exercises
305(4)
7 PROPERTIES OF EXPECTATION
309(86)
7.1 Introduction
309(1)
7.2 Expectation of Sums of Random Variables
310(15)
7.3 Covariance, Variance of Sums, and Correlations
325(10)
7.4 Conditional Expectation
335(15)
7.4.1 Definitions
335(2)
7.4.2 Computing Expectations by Conditioning
337(7)
7.4.3 Computing Probabilities by Conditioning
344(4)
7.4.4 Conditional Variance
348(2)
7.5 Conditional Expectation and Prediction
350(5)
7.6 Moment Generating Functions
355(10)
7.6.1 Joint Moment Generating Functions
364(1)
7.7 Additional Properties of Normal Random Variables
365(3)
7.7.1 The Multivariate Normal Distribution
365(1)
7.7.2 The Joint Distribution of the Sample Mean and Sample Variance
366(2)
7.8 General Definition of Expectation(*)
368(2)
Summary
370(2)
Problems
372(12)
Theoretical Exercises
384(8)
Self-Test Problems and Exercises
392(3)
8 LIMIT THEOREMS
395(33)
8.1 Introduction
395(1)
8.2 Chebyshev's Inequality and the Weak Law of Large Numbers
395(4)
8.3 The Central Limit Theorem
399(8)
8.4 The Strong Law of Large Numbers
407(5)
8.5 Other Inequalities
412(6)
8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson
418(3)
Summary
421(1)
Problems
421(3)
Theoretical Exercises
424(2)
Self-Test Problems and Exercises
426(2)
9 ADDITIONAL TOPICS IN PROBABILITY
428(24)
9.1 The Poisson Process
428(3)
9.2 Markov Chains
431(5)
9.3 Surprise, Uncertainty, and Entropy
436(5)
9.4 Coding Theory and Entropy
441(1)
Summary
441(7)
Theoretical Exercises and Problems
448(2)
Self-Test Problems and Exercises
450(1)
References
450(2)
10 SIMULATION
452(21)
10.1 Introduction
452(3)
10.2 General Techniques for Simulating Continuous Random Variables
455(7)
10.2.1 The Inverse Transformation Method
455(1)
10.2.2 The Rejection Method
456(6)
10.3 Simulating from Discrete Distributions
462(2)
10.4 Variance Reduction Techniques
464(4)
10.4.1 Use of Antithetic Variables
465(1)
10.4.2 Variance Reduction by Conditioning
466(2)
10.4.3 Control Variates
468(1)
Summary
468(1)
Problems
469(2)
Self-Test Problems and Exercises
472(1)
References
472(1)
Appendix A ANSWERS TO SELECTED PROBLEMS473(4)
Appendix B SOLUTIONS TO SELF-TEST PROBLEMS AND EXERCISES477(36)
INDEX513

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