Preface 

vi  


1  (23) 


1  (1) 

The Basic Principle of Counting 


2  (1) 


3  (2) 


5  (5) 


10  (2) 

The Number of Integer Solutions of Equations* 


12  (12) 


15  (1) 


15  (7) 


22  (1) 

SelfTest Problems and Exercises 


22  (2) 


24  (40) 


24  (1) 


24  (4) 


28  (3) 


31  (4) 

Sample Spaces Having Equally Likely Outcomes 


35  (12) 

Probability As a Continuous Set Function* 


47  (4) 

Probability As a Measure of Belief 


51  (13) 


52  (1) 


53  (6) 


59  (2) 

SelfTest Problems and Exercises 


61  (3) 

Conditional Probability and Independence 


64  (58) 


64  (1) 

Conditional Probabilities 


64  (5) 


69  (14) 


83  (13) 


96  (26) 


103  (1) 


104  (11) 


115  (4) 

SelfTest Problems and Exercises 


119  (3) 


122  (65) 


122  (5) 

Discrete Random Variables 


127  (3) 


130  (3) 

Expectation of a Function of a Random Variable 


133  (4) 


137  (2) 

The Bernoulli and Binomial Random Variables 


139  (10) 

Properties of Binomial Random Variables 


144  (3) 

Computing the Binomial Distribution Function 


147  (2) 

The Poisson Random Variable 


149  (9) 

Computing the Poisson Distribution Function 


157  (1) 

Other Discrete Probability Distribution 


158  (8) 

The Geometric Random Variable 


158  (2) 

The Negative Binomial Random Variable 


160  (2) 

The Hypergeometric Random Variable 


162  (4) 

The Zeta (or Zipf) distribution 


166  (1) 

Properties of the Cumulative Distribution Function 


166  (21) 


169  (2) 


171  (9) 


180  (4) 

SelfTest Problems and Exercises 


184  (3) 

Continuous Random Variables 


187  (52) 


187  (3) 

Expectation and Variance of Continuous Random Variables 


190  (5) 

The Uniform Random Variable 


195  (4) 


199  (11) 

The Normal Approximation to the Binomial Distribution 


206  (4) 

Exponential Random Variables 


210  (7) 


215  (2) 

Other Continuous Distributions 


217  (6) 


217  (3) 


220  (1) 


220  (1) 


221  (2) 

The Distribution of a Function of a Random Variable 


223  (16) 


225  (3) 


228  (4) 


232  (3) 

SelfTest Problems and Exercises 


235  (4) 

Jointly Distributed Random Variables 


239  (65) 

Joint Distribution Functions 


239  (9) 

Independent Random Variables 


248  (12) 

Sums of Independent Random Variables 


260  (8) 

Conditional Distributions: Discrete Case 


268  (2) 

Conditional Distributions: Continuous Case 


270  (3) 


273  (4) 

Joint Probability Distribution of Functions of Random Variables 


277  (8) 

Exchangeable Random Variables* 


285  (19) 


288  (2) 


290  (6) 


296  (3) 

SelfTest Problems and Exercises 


299  (5) 

Properties of Expectations 


304  (96) 


304  (1) 

Expectation of Sums of Random Variables 


305  (22) 

Obtaining Bounds from Expectations via the Probabilistic Method* 


321  (3) 

The MaximumMinimums Identity* 


324  (3) 

Covariance, Variance of Sums, and Correlations 


327  (13) 


340  (16) 


340  (3) 

Computing Expectations by Conditioning 


343  (7) 

Computing Probabilities by Conditioning 


350  (4) 


354  (2) 

Conditional Expectation and Prediction 


356  (5) 

Moment Generating Functions 


361  (12) 

Joint Moment Generating Functions 


371  (2) 

Additional Properties of Normal Random Variables 


373  (2) 

The Multivariate Normal Distribution 


373  (1) 

The Joint Distribution of the Sample Mean and Sample Variance 


374  (1) 

General Definition of Expectation* 


375  (25) 


377  (2) 


379  (10) 


389  (8) 

SelfTest Problems and Exercises 


397  (3) 


400  (32) 


400  (1) 

Chebyshev's Inequality and the Weak Law of Large Numbers 


400  (3) 

The Central Limit Theorem 


403  (9) 

The Strong Law of Large Numbers 


412  (5) 


417  (7) 

Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson 


424  (8) 


426  (1) 


427  (2) 


429  (1) 

SelfTest Problems and Exercises 


430  (2) 

Additional Topics in Probability 


432  (23) 


432  (3) 


435  (5) 

Surprise, Uncertainty, and Entropy 


440  (5) 

Coding Theory and Entropy 


445  (10) 


451  (1) 

Theoretical Exercises and Problems 


452  (2) 

SelfTest Problems and Exercises 


454  (1) 


454  (1) 


455  (20) 


455  (3) 

General Techniques for Simulating Continuous Random Variables 


458  (7) 

The Inverse Transformation Method 


458  (1) 


459  (6) 

Simulating from Discrete Distributions 


465  (2) 

Variance Reduction Techniques 


467  (8) 

Use of Antithetic Variables 


468  (1) 

Variance Reduction by Conditioning 


468  (2) 


470  (1) 


471  (1) 


471  (3) 

SelfTest Problems and Exercises 


474  (1) 


474  (1) 
Appendix A Answers to Selected Problems 

475  (3) 
Appendix B Solutions to SelfTest Problems and Exercises 

478  (41) 
Index 

519  