Preface 

vii  


1  (23) 


1  (1) 

The Basic Principle of Counting 


2  (1) 


3  (3) 


6  (4) 


10  (2) 

The Number of Integer Solutions of Equations* 


12  (12) 


15  (1) 


16  (3) 


19  (3) 

SelfTest Problems and Exercises 


22  (2) 


24  (42) 


24  (1) 


24  (5) 


29  (2) 


31  (6) 

Sample Spaces Having Equally Likely Outcomes 


37  (12) 

Probability as a Continuous Set Function* 


49  (4) 

Probability as a Measure of Belief 


53  (13) 


54  (1) 


55  (6) 


61  (2) 

SelfTest Problems and Exercises 


63  (3) 

Conditional Probability and Independence 


66  (66) 


66  (1) 

Conditional Probabilities 


66  (6) 


72  (15) 


87  (14) 


101  (31) 


110  (1) 


111  (13) 


124  (4) 

SelfTest Problems and Exercises 


128  (4) 


132  (73) 


132  (6) 

Discrete Random Variables 


138  (2) 


140  (4) 

Expectation of a Function of a Random Variable 


144  (4) 


148  (2) 

The Bernoulli and Binomial Random Variables 


150  (10) 

Properties of Binomial Random Variables 


155  (3) 

Computing the Binomial Distribution Function 


158  (2) 

The Poisson Random Variable 


160  (13) 

Computing the Poisson Distribution Function 


173  (1) 

Other Discrete Probability Distributions 


173  (10) 

The Geometric Random Variable 


173  (2) 

The Negative Binomial Random Variable 


175  (3) 

The Hypergeometric Random Variable 


178  (4) 

The Zeta (or Zipf) Distribution 


182  (1) 

Properties of the Cumulative Distribution Function 


183  (22) 


185  (2) 


187  (10) 


197  (4) 

SelfTest Problems and Exercises 


201  (4) 

Continuous Random Variables 


205  (53) 


205  (4) 

Expectation and Variance of Continuous Random Variables 


209  (5) 

The Uniform Random Variable 


214  (4) 


218  (12) 

The Normal Approximation to the Binomial Distribution 


225  (5) 

Exponential Random Variables 


230  (7) 


234  (3) 

Other Continuous Distributions 


237  (5) 


237  (2) 


239  (1) 


239  (1) 


240  (2) 

The Distribution of a Function of a Random Variable 


242  (16) 


244  (3) 


247  (4) 


251  (3) 

SelfTest Problems and Exercises 


254  (4) 

Jointly Distributed Random Variables 


258  (69) 

Joint Distribution Functions 


258  (9) 

Independent Random Variables 


267  (13) 

Sums of Independent Random Variables 


280  (8) 

Conditional Distributions: Discrete Case 


288  (3) 

Conditional Distributions: Continuous Case 


291  (5) 


296  (4) 

Joint Probability Distribution of Functions of Random Variables 


300  (8) 

Exchangeable Random Variables* 


308  (19) 


311  (2) 


313  (6) 


319  (4) 

SelfTest Problems and Exercises 


323  (4) 

Properties of Expectation 


327  (103) 


327  (1) 

Expectation of Sums of Random Variables 


328  (19) 

Obtaining Bounds from Expectations via the Probabilistic Method* 


342  (2) 

The MaximumMinimums Identity* 


344  (3) 

Moments of the Number of Events that Occur 


347  (8) 

Covariance, Variance of Sums, and Correlations 


355  (10) 


365  (17) 


365  (2) 

Computing Expectations by Conditioning 


367  (9) 

Computing Probabilities by Conditioning 


376  (4) 


380  (2) 

Conditional Expectation and Prediction 


382  (5) 

Moment Generating Functions 


387  (12) 

Joint Moment Generating Functions 


397  (2) 

Additional Properties of Normal Random Variables 


399  (5) 

The Multivariate Normal Distribution 


399  (3) 

The Joint Distribution of the Sample Mean and Sample Variance 


402  (2) 

General Definition of Expectation 


404  (26) 


405  (3) 


408  (10) 


418  (8) 

SelfTest Problems and Exercises 


426  (4) 


430  (33) 


430  (1) 

Chebyshev's Inequality and the Weak Law of Large Numbers 


430  (4) 

The Central Limit Theorem 


434  (9) 

The Strong Law of Large Numbers 


443  (2) 


445  (9) 

Bounding The Error Probability 


454  (9) 


456  (1) 


457  (2) 


459  (2) 

SelfTest Problems and Exercises 


461  (2) 

Additional Topics in Probability 


463  (24) 


463  (3) 


466  (6) 

Surprise, Uncertainty, and Entropy 


472  (4) 

Coding Theory and Entropy 


476  (11) 


483  (1) 


484  (1) 

SelfTest Problems and Exercises 


485  (2) 


487  (74) 


487  (3) 

General Techniques for Simulating Continuous Random Variables 


490  (7) 

The Inverse Transformation Method 


490  (1) 


491  (6) 

Simulating from Discrete Distributions 


497  (2) 

Variance Reduction Techniques 


499  (9) 

Use of Antithetic Variables 


500  (1) 

Variance Reduction by Conditioning 


501  (2) 


503  (1) 


503  (1) 


504  (2) 

SelfTest Problems and Exercises 


506  (2) 



A Answers to Selected Problems 


508  (3) 

B Solutions to SelfTest Problems and Exercises 


511  (50) 
Index 

561  