**Introduction **

This text is intended as an introduction to elementary probability theory and stochastic processes. It is particularly well suited for those wanting to see how probabilitytheorycanbeappliedtothestudyofphenomenainﬁeldssuchasengineering, computer science, management science, the physical and social sciences, and operations research. Itisgenerallyfeltthattherearetwoapproachestothestudyofprobabilitytheory. One approach is heuristic and no rigorous and attempts to develop in the student an intuitive feel for the subject that enables him or her to “think probabilistically.” The other approach attempts a rigorous development of probability by using the tools of measure theory. It is the ﬁrst approach that is employed in this text. However, because it is extremely important in both understanding and applying probability theory to be able to “think probabilistically,” this text should also be useful to students interested primarily in the second approach.

The tenth edition includes new text material, examples, and exercises chosen not only for their inherent interest and applicability but also for their usefulness in strengthening the reader’s probabilistic knowledge and intuition. The new text material includes Section 2.7, which builds on the inclusion/exclusion identity to ﬁnd the distribution of the number of events that occur; and Section 3.6.6 on left skip free random walks, which can be used to model the fortunes of an investor (or gambler)who always invests 1 and then receives an on negative integral return. Section4.2hasadditionalmaterialonMarkovchainsthatshowshowtomodifya givenchainwhentryingtodeterminesuchthingsastheprobabilitythatthechain ever enters a given class of states by some time, or the conditional distribution of the state at some time given that the class has never been entered. A new remark inSection7.2showsthatresultsfromtheclassicalinsuranceruinmodelalsohold in other important ruin models. There is new material on exponential queueing models, including, in Section 2.2, a determination of the mean and variance of the number of lost customers in a busy period of a ﬁnite capacity queue, as well a the new Section 8.3.3 on birth and death queueing models. Section 11.8.2 gives a new approach that can be used to simulate the exact stationary distribution of a Markov chain that satisﬁes a certain property. Amongthenewlyaddedexamplesare1.11,whichisconcernedwithamultiple player gambling problem; 3.20, which ﬁnds the variance in the matching rounds problem; 3.30, which deals with the characteristics of a random selection from a population; and 4.25, which deals with the stationary distribution of a Markov chain

**Contents**

1 Introduction to Probability Theory

1 Introduction to Probability Theory

1.1 Introduction

1.2 Sample Space and Events

1.3 Probabilities Deﬁned on Events

1.4 Conditional Probabilities

7 1.5 Independent Events

1.6 Bayes’ Formula

Exercises

References

**2 Random Variables**

2.1 Random Variables

2.2 Discrete Random Variables

2.2.1 The Bernoulli Random Variable

2.2.2 The Binomial Random Variable

2.2.3 The Geometric Random Variable

2.2.4 The Poisson Random Variable

2.3 Continuous Random Variables

2.3.1 The Uniform Random Variable

2.3.2 Exponential Random Variables

2.3.3 Gamma Random Variables

2.3.4 Normal Random Variables

2.4 Expectation of a Random Variable

2.4.1 The Discrete Case

2.4.2 The Continuous Case

2.4.3 Expectation of a Function of a Random Variable

2.5 Jointly Distributed Random Variables

2.5.1 Joint Distribution Functions

2.5.2 Independent Random Variables

2.5.3 Covariance and Variance of Sums of Random Variables

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

2.7 The Distribution of the Number of Events that Occur

2.8 Limit Theorems

2.9 Stochastic

Processes

Exercises

References

**3 Conditional Probability and Conditional Expectation **

3.1 Introduction

3.2 The Discrete Case

3.3 The Continuous Case

3.4 Computing Expectations by Conditioning

3.4.1 Computing Variances by Conditioning

3.5 Computing Probabilities by Conditioning

3.6 Some Applications

3.6.1 A List Model

3.6.2 A Random Graph

3.6.3 Uniform Priors, Polya’s Urn Model, and Bose–Einstein Statistics

3.6.4 Mean Time for Patterns

3.6.5 The k-Record Values of Discrete Random Variables

3.6.6 Left Skip Free Random Walks

3.7 An Identity for Compound Random Variables

3.7.1 Poisson Compounding Distribution

3.7.2 Binomial Compounding Distribution

3.7.3 A Compounding Distribution Related to the Negative Binomial

Exercises

**4 Markov Chains**

4.1 Introduction

4.2 Chapman–Kolmogorov Equations

4.3 Classiﬁcation of States

4.4 Limiting Probabilities

4.5 Some Applications

4.5.1 The Gambler’s Ruin Problem

4.5.2 A Model for Algorithmic Efﬁciency

4.5.3 Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisﬁability Problem

4.6 Mean Time Spent in Transient States

4.7 Branching Processes

Time Reversible Markov Chains

4.9 Markov Chain Monte Carlo Methods

4.10 Markov Decision Processes

4.11 Hidden Markov Chains

4.11.1 Predicting the States

Exercises

References

**5 The Exponential Distribution and the Poisson Process**

5.1 Introduction

5.2 The Exponential Distribution

5.2.1 Deﬁnition

5.2.2 Properties of the Exponential Distribution

5.2.3 Further Properties of the Exponential Distribution

5.2.4 Convolutions of Exponential Random Variables

5.3 The Poisson Process

5.3.1 Counting Processes

5.3.2 Deﬁnition of the Poisson Process

5.3.3 Interarrival and Waiting Time Distributions

5.3.4 Further Properties of Poisson Processes

5.3.5 Conditional Distribution of the Arrival Times

5.3.6 Estimating Software Reliability

5.4 Generalizations of the Poisson Process

5.4.1 Nonhomogeneous Poisson Process

5.4.2 Compound Poisson Process

5.4.3 Conditional or Mixed Poisson Processes

Exercises

References

**6 Continuous-Time Markov Chains**

6.1 Introduction

6.2 Continuous-Time Markov Chains

6.3 Birth and Death Processes

6.4 The Transition Probability Function Pij(t)

6.5 Limiting Probabilities 390 6.6 Time Reversibility

6.7 Uniformization

6.8 Computing the Transition Probabilities

Exercises

References

**7 Renewal Theory and Its Applications**

7.1 Introduction

7.2 Distribution of N(t)

7.3 Limit Theorems and Their Applications

7.4 Renewal Reward Processes

7.5 Regenerative Processes

7.5.1 Alternating Renewal Processes

7.6 Semi-Markov Processes

7.7 The Inspection Paradox

7.8 Computing the Renewal Function

7.9 Applications to Patterns

7.9.1 Patterns of Discrete Random Variables

7.9.2 The Expected Time to a Maximal Run of Distinct Values

7.9.3 Increasing Runs of Continuous Random Variables

7.10 The Insurance Ruin Problem

Exercises

References

**8 Queueing Theory **

8.1 Introduction

8.2 Preliminaries

8.2.1 Cost Equations

8.2.2 Steady-State Probabilities

8.3 Exponential Models

8.3.1 A Single-Server Exponential Queueing System

8.3.2 A Single-Server Exponential Queueing System Having Finite Capacity

8.3.3 Birth and Death Queueing Models

8.3.4 A Shoe Shine Shop

8.3.5 A Queueing System with Bulk Service 524 8.4 Network of Queues

8.4.1 Open Systems

8.4.2 Closed Systems

8.5 The System M/G/1

8.5.1 Preliminaries: Work and another Cost Identity

8.5.2 Application of Work to M/G/1

8.5.3 Busy Periods 540 8.6 Variations on the M/G/1

8.6.1 The M/G/1 with Random-Sized Batch Arrivals

8.6.2 Priority Queues

8.6.3 An M/G/1 Optimization Example

8.6.4 The M/G/1 Queue with Server Breakdown

8.7 The Model G/M/1

8.7.1 The G/M/1 Busy and Idle Periods

8.8 A Finite Source Model

8.9 Multiserver Queues

8.9.1 Erlang’s Loss System

8.9.2 The M/M/k Queue

8.9.3 The G/M/k Queue

8.9.4 The M/G/k Queue

Exercises

References

**9 Reliability Theory**

9.1 Introduction

9.2 Structure Functions

9.2.1 Minimal Path and Minimal Cut Sets

9.3 Reliability of Systems of Independent Components

9.4 Bounds on the Reliability Function

9.4.1 Method of Inclusion and Exclusion

9.4.2 Second Method for Obtaining Bounds on r(p)

9.5 System Life as a Function of Component Lives

9.6 Expected System Lifetime

9.6.1 An Upper Bound on the Expected Life of a Parallel System

9.7 Systems with Repair

9.7.1 A Series Model with Suspended Animation

Exercises

References

**10 Brownian Motion and Stationary Processes**

10.1 Brownian Motion

10.2 Hitting Times, Maximum Variable, and the Gambler’s Ruin Problem

10.3 Variations on Brownian motion

10.3.1 Brownian Motion with Drift

10.3.2 Geometric Brownian motion

10.4 Pricing Stock Options

10.4.1 An Example in Options Pricing

10.4.2 The Arbitrage Theorem

10.4.3 The Black-Scholes Option Pricing Formula

10.5 White Noise

10.6 Gaussian Processes

10.7 Stationary and Weakly Stationary Processes

10.8 Harmonic Analysis of Weakly Stationary Processes

Exercises

References 665

**11 Simulation **

11.1 Introduction

11.2 General Techniques for Simulating Continuous Random Variables

11.2.1 The Inverse Transformation Method

11.2.2 The Rejection Method

11.2.3 The Hazard Rate Method

11.3 Special Techniques for Simulating Continuous Random Variables

11.3.1 The Normal Distribution

11.3.2 The Gamma Distribution

11.3.3 The Chi-Squared Distribution

11.3.4 The Beta (n,m) Distribution

11.3.5 The Exponential Distribution—The Von Neumann Algorithm

11.4 Simulating from Discrete Distributions

11.4.1 The Alias Method

11.5 Stochastic Processes

11.5.1 Simulating a Nonhomogeneous Poisson Process

11.5.2 Simulating a Two-Dimensional Poisson Process

11.6 Variance Reduction Techniques

11.6.1 Use of Antithetic Variables

11.6.2 Variance Reduction by Conditioning

11.6.3 Control Variates

11.6.4 Importance Sampling

11.7 Determining the Number of Runs

11.8 Generating from the Stationary Distribution of a Markov Chain

11.8.1 Coupling from the Past

11.8.2 Another Approach

Exercises

References

Appendix: Solutions to Starred Exercises 735

Index

Tags: #Introduction to Probability Models 10th Edition #Introduction to Probability Models Ross 10th edition pdf #Introduction to Probability Models Sheldon M. Ross 10th edition (2017) pdf