应用随机过程:概率模型导论(英文版·第12版) [美]谢尔登·M.罗斯 著 无 译 专业科技 文轩网

1 Introduction to Probability Theory 1 1.1  Introduction  1 1.2  Sample Space and Events  1 1.3  Probabilities Defined on Events  3 1.4  Conditional Probabilities  6 1.5  Independent Events  9 1.6  Bayes’ Formula  11 1.7  Probability Is a Continuous Event Function  14 Exercises  15 References  21 2  Random Variables  23 2.1  Random Variables  23 2.2  Discrete Random Variables  27 2.2.1  The Bernoulli Random Variable  28 2.2.2  The Binomial Random Variable  28 2.2.3  The Geometric Random Variable  30 2.2.4  The Poisson Random Variable  31 2.3  Continuous Random Variables  32 2.3.1  The Uniform Random Variable  33 2.3.2  Exponential Random Variables  35 2.3.3  Gamma Random Variables  35 2.3.4  Normal Random Variables  35 2.4  Expectation of a Random Variable  37 2.4.1  The Discrete Case  37 2.4.2  The Continuous Case  39 2.4.3  Expectation of a Function of a Random Variable  41 2.5  Jointly Distributed Random Variables  44 2.5.1  Joint Distribution Functions  44 2.5.2  Independent Random Variables  49 2.5.3  Covariance and Variance of Sums of Random Variables 50 Properties of Covariance  52 2.5.4  Joint Probability Distribution of Functions of Random Variables  59 2.6  Moment Generating Functions  62 2.6.1  The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population  70 2.7  Limit Theorems  73 2.8  Proof of the Strong Law of Large Numbers  79 2.9  Stochastic Processes  84 Exercises  86 References  99 3  Conditional Probability and Conditional Expectation  101 3.1  Introduction  101 3.2  The Discrete Case  101 3.3  The Continuous Case  104 3.4  Computing Expectations by Conditioning  108 3.4.1  Computing Variances by Conditioning  120 3.5  Computing Probabilities by Conditioning  124 3.6  Some Applications  143 3.6.1  A List Model  143 3.6.2  A Random Graph  145 3.6.3  Uniform Priors, Polya’s Urn Model, and Bose–Einstein Statistics  152 3.6.4  Mean Time for Patterns  156 3.6.5  The k-Record Values of Discrete Random Variables  159 3.6.6  Left Skip Free Random Walks  162 3.7  An Identity for Compound Random Variables  168 3.7.1  Poisson Compounding Distribution  171 3.7.2  Binomial Compounding Distribution  172 3.7.3  A Compounding Distribution Related to the Negative Binomial  173 Exercises  174 4  Markov Chains  193 4.1  Introduction  193 4.2  Chapman–Kolmogorov Equations  197 4.3  Classification of States  205 4.4  Long-Run Proportions and Limiting Probabilities  215 4.4.1  Limiting Probabilities  232 4.5  Some Applications  233 4.5.1  The Gambler’s Ruin Problem  233 4.5.2  A Model for Algorithmic Efficiency  237 4.5.3  Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem  239 4.6  Mean Time Spent in Transient States  245 4.7  Branching Processes 2475.2.2 Properties of the Exponential Distribution  295 4.8  Time Reversible Markov Chains  251 4.9  Markov Chain Monte Carlo Methods  261 4.10  Markov Decision Processes  265 4.11  Hidden Markov Chains  269 4.11.1  Predicting the States  273 Exercises  275 References  291 5  The Exponential Distribution and the Poisson Process  293 5.1  Introduction  293 5.2  The Exponential Distribution  293 5.2.1  Definition  293 5.2.2  Properties of the Exponential Distribution  295 5.2.3  Further Properties of the Exponential Distribution  302 5.2.4  Convolutions of Exponential Random Variables  309 5.2.5  The Dirichlet Distribution  313 5.3  The Poisson Process  314 5.3.1  Counting Processes  314 5.3.2  Definition of the Poisson Process  316 5.3.3  Further Properties of Poisson Processes  320 5.3.4  Conditional Distribution of the Arrival Times  326 5.3.5  Estimating Software Reliability  336 5.4  Generalizations of the Poisson Process  339 5.4.1  Nonhomogeneous Poisson Process  339 5.4.2  Compound Poisson Process  346 Examples  of Compound Poisson Processes  346 5.4.3  Conditional or Mixed Poisson Processes  351 5.5  Random Intensity Functions and Hawkes Processes  353 Exercises  357 References  374 6  Continuous-Time Markov Chains  375 6.1  Introduction  375 6.2  Continuous-Time Markov Chains  375 6.3  Birth and Death Processes  377 6.4  The Transition Probability Function Pi j (t)  384 6.5  Limiting Probabilities  394 6.6  Time Reversibility  401 6.7  The Reversed Chain  409 6.8  Uniformization  414 6.9  Computing the Transition Probabilities  418 Exercises  420 References  429 7  Renewal Theory and Its Applications  431 7.1  Introduction  431 7.2  Distribution of N (t)  432 7.3  Limit Theorems and Their Applications  436 7.4  Renewal Reward Processes  450 7.5  Regenerative Processes  461 7.5.1  Alternating Renewal Processes  464 7.6  Semi-Markov Processes  470 7.7  The Inspection Paradox  473 7.8  Computing the Renewal Function  476 7.9  Applications to Patterns  479 7.9.1  Patterns of Discrete Random Variables  479 7.9.2  The Expected Time to a Maximal Run of Distinct Values  486 7.9.3  Increasing Runs of Continuous Random Variables  488 7.10  The Insurance Ruin Problem  489 Exercises  495 References  506 8  Queueing Theory  507 8.1  Introduction  507 8.2  Preliminaries  508 8.2.1  Cost Equations  508 8.2.2  Steady-State Probabilities  509 8.3  Exponential Models  512 8.3.1  A Single-Server Exponential Queueing System  512 8.3.2  A Single-Server Exponential Queueing System Having Finite Capacity  522 8.3.3  Birth and Death Queueing Models  527 8.3.4  A Shoe Shine Shop  534 8.3.5  Queueing Systems with Bulk Service  536 8.4  Network of Queues  540 8.4.1  Open Systems  540 8.4.2  Closed Systems  544 8.5  The System M/G/1  549 8.5.1  Preliminaries: Work and Another Cost Identity  549 8.5.2  Application of Work to M/G/1  550 8.5.3  Busy Periods  552 8.6  Variations on the M/G/1  554 8.6.1  The M/G/1 with Random-Sized Batch Arrivals  554 8.6.2  Priority Queues  555 8.6.3  An M/G/1 Optimization Example  558 8.6.4  The M/G/1 Queue with Server Breakdown  562 8.7  The Model G/M/1  565 8.7.1  The G/M/1 Busy and Idle Periods  569 8.8  A Finite Source Model  570 8.9  Multiserver Queues  573 8.9.1  Erlang’s Loss System  574 8.9.2  The M/M/k Queue  575 8.9.3  The G/M/k Queue  575 8.9.4  The M/G/k Queue  577 Exercises  578 9  Reliability Theory  591 9.1  Introduction  591 9.2  Structure Functions  591 9.2.1  Minimal Path and Minimal Cut Sets  594 9.3  Reliability of Systems of Independent Components  597 9.4  Bounds on the Reliability Function  601 9.4.1  Method of Inclusion and Exclusion  602 9.4.2  Second Method for Obtaining Bounds on r(p)  610 9.5  System Life as a Function of Component Lives  613 9.6  Expected System Lifetime  620 9.6.1  An Upper Bound on the Expected Life of a Parallel System  623 9.7  Systems with Repair  625 9.7.1  A Series Model with Suspended Animation  630 Exercises  632 References  638 10  Brownian Motion and Stationary Processes  639 10.1  Brownian Motion  639 10.2  Hitting Times, Maximum Variable, and the Gambler’s Ruin Problem  643 10.3  Variations on Brownian Motion  644 10.3.1  Brownian Motion with Drift  644 10.3.2  Geometric Brownian Motion  644 10.4  Pricing Stock Options  646 10.4.1  An Example in Options Pricing  646 10.4.2  The Arbitrage Theorem  648 10.4.3  The Black–Scholes Option Pricing Formula  651 10.5  The Maximum of Brownian Motion with Drift  656 10.6  White Noise  661 10.7  Gaussian Processes  663 10.8  Stationary and Weakly Stationary Processes  665 10.9  Harmonic Analysis of Weakly Stationary Processes  670 Exercises  672 References  677 11  Simulation  679 11.1  Introduction  679 11.2  General Techniques for Simulating Continuous Random Variables  683 11.2.1  The Inverse Transformation Method  683 11.2.2  The Rejection Method  684 11.2.3  The Hazard Rate Method  688 11.3  Spe Techniques for Simulating Continuous Random Variables  691 11.3.1  The Normal Distribution  691 11.3.2  The Gamma Distribution  694 11.3.3  The Chi-Squared Distribution  695 11.3.4  The Beta (n, m) Distribution  695 11.3.5  The Exponential Distribution—The Von Neumann Algorithm  696 11.4  Simulating from Discrete Distributions  698 11.4.1  The Alias Method  701 11.5  Stochastic Processes  705 11.5.1  Simulating a Nonhomogeneous Poisson Process  706 11.5.2  Simulating a Two-Dimensional Poisson Process  712 11.6  Variance Reduction Techniques  715 11.6.1  Use of Antithetic Variables  716 11.6.2  Variance Reduction by Conditioning  719 11.6.3  Control Variates  723 11.6.4  Importance Sampling  725 11.7  Determining the Number of Runs  730 11.8  Generating from the Stationary Distribution of a Markov Chain  731 11.8.1  Coupling from the Past  731 11.8.2  Another Approach  733 Exercises  734 References  741 12  Coupling  743 12.1  A Brief Introduction  743 12.2  Coupling and Stochastic Order Relations  743 12.3  Stochastic Ordering of Stochastic Processes  746 12.4  Maximum Couplings, Total Variation Distance, and the Coupling Identity  749 12.5  Applications of the Coupling Identity  752 12.5.1  Applications to Markov Chains  752 12.6  Coupling and Stochastic Optimization  758 12.7  Chen–Stein Poisson Approximation Bounds  762 Exercises  769 Solutions  to Starred Exercises  773 Index  817