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## Über dieses Buch

This textbook explores probability and stochastic processes at a level that does not require any prior knowledge except basic calculus. It presents the fundamental concepts in a step-by-step manner, and offers remarks and warnings for deeper insights. The chapters include basic examples, which are revisited as the new concepts are introduced. To aid learning, figures and diagrams are used to help readers grasp the concepts, and the solutions to the exercises and problems. Further, a table format is also used where relevant for better comparison of the ideas and formulae. The first part of the book introduces readers to the essentials of probability, including combinatorial analysis, conditional probability, and discrete and continuous random variable. The second part then covers fundamental stochastic processes, including point, counting, renewal and regenerative processes, the Poisson process, Markov chains, queuing models and reliability theory. Primarily intended for undergraduate engineering students, it is also useful for graduate-level students wanting to refresh their knowledge of the basics of probability and stochastic processes.

## Inhaltsverzeichnis

### Chapter 1. Combinatorial Analysis

Abstract
In this chapter, the basics of combinatorial analysis including the basic principle of counting, generalized basic principle of counting, permutation with repetition and without repetition, and combination with sequence consideration and without sequence consideration have been provided. The distribution of a set of elements into distinct groups and into indistinct groups has also been defined as a special application of the combination without sequence consideration. Some exercises throughout the chapter and problems at the end of the chapter have been provided to clarify the basic concepts. Some problems have also been presented to make connections to some well-known engineering problems such as traveling salesperson (salesman) problem, 0–1 knapsack problem, and job scheduling problem.
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### Chapter 2. Basic Concepts, Axioms and Operations in Probability

Abstract
In this chapter, the definitions of the basic concepts in probability including the sample space, single event, event, mutually exclusive events, probability, and three axioms of probability have been provided. The basic operations in probability theory including the intersection, union, and complement have been given in comparison to the basic operations in set theory. The exercises throughout the chapter, and the problems at the end of the chapter elucidated the basic concepts in probability. In problems section, Venn diagrams have been used for the solutions, where relevant; and the solution of two well-known problems in probability, i.e. birthday problem, and straight in a poker hand problem was also presented.
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### Chapter 3. Conditional Probability, Bayes’ Formula, Independent Events

Abstract
In this chapter, three important topics, i.e. the conditional probability, Bayes’s formula, and the independence of two and three events have been introduced, which are crucial not only for the basics of probability, but also for the basics of stochastic processes. Some illustrative examples and problems have been presented, and most of the solutions have been supported by using the tree diagrams for a better comprehension.
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### Chapter 4. Introduction to Random Variables

Abstract
In this chapter, the basic concepts for both discrete and continuous random variables were introduced. The definition of a random variable, a discrete random variable, and a continuous random variable, and the formulae of the probability mass function, probability density function, cumulative distribution function, complementary cumulative distribution function (tail function), expected value of a random variable, expected value of a function of a random variable, and the variance and standard deviation of a random variable have been provided. In addition to the solutions of some typical examples and problems for the discrete and continuous random variables, one well-known problem, i.e. newsboy (newsvendor) problem has also been presented.
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### Chapter 5. Discrete Random Variables

Abstract
This chapter introduced some special discrete random variables, i.e. binomial random variable and Bernoulli random variable, Poisson random variable, negative binomial random variable, geometric random variable, and hypergeometric random variable. The formulae for the basic parameters of these special discrete random variables have been provided. In addition to the typical examples and problems such as the number of the typographical errors on a book page, and the ball withdrawal from an urn without replacement, one special problem that is related to acceptance sampling in quality control has also been presented.
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### Chapter 6. Continuous Random Variables

Abstract
This chapter introduced special continuous random variables including uniform random variable, exponential random variable, normal random variable, standard normal random variable, and other continuous random variables including gamma random variable, lognormal random variable, and Weibull random variable. The standard normal random variable approximation to a binomial random variable has also been provided. The formulae of the basic parameters for each continuous random variable have been given, and several typical examples and problems have been presented for each continuous random variable.
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### Chapter 7. Other Selected Topics in Basic Probability

Abstract
In the first part of this chapter, the basic terms related to the jointly distributed random variables including the joint probability mass function and joint probability density function, sums of independent random variables, convolution of random variables, order statistics of continuous random variables, covariance and correlation of random variables, and expected value and variance of sum of random variables have been explained. The second part of this chapter has been devoted to the conditional distribution, conditional expected value, conditional variance, expected value by conditioning, and variance by conditioning. In the third part of the chapter, the moment generating function and characteristic function have been defined, and a list of moment generating functions has been provided for special random variables. The last part of the chapter included the limit theorems in probability including Strong Law of Large Numbers and Central Limit Theorem. The concepts have been illustrated with several examples and problems.
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### Chapter 8. A Brief Introduction to Stochastic Processes

Abstract
This chapter was a very brief introduction to the basic concepts of the stochastic processes including the definition of a stochastic process, a discrete-time and continuous-time stochastic process, state space, i.i.d. stochastic process, stopping time, and the hitting time of a process to a state. Some typical examples and problems including the daily stock prices of a company, and the evolution of the population of a country have been provided to clarify the basic concepts.
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### Chapter 9. A Brief Introduction to Point Process, Counting Process, Renewal Process, Regenerative Process, Poisson Process

Abstract
This chapter started with the definition of point process as a basic stochastic process, continued with the definitions of the arrival time and the interarrival time; and based on the definition of point process, counting process has been defined. The definition of renewal process as a special counting process, and the definition of Poisson process as a special renewal process have been provided. Poisson process has been classified as homogeneous and nonhomogeneous Poisson process, and the basic properties of a homogeneous Poisson process including the stationary and independent increments, exponential random variable as the interarrival time distribution, and gamma random variable as the arrival time distribution have been explained. In addition to the typical example of the arrival of the customers; other illustrative examples including the breakdown of the machines, time of the earthquakes, and the two-server systems have also been presented, and solved by using the basic formulae and the schematic representations.
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### Chapter 10. Poisson Process

Abstract
This chapter started with two formal definitions of a homogeneous Poisson process and a nonhomogeneous Poisson process, and the typical example of the arrivals of the customers. Some additional properties of a homogeneous Poisson process including partitioning a homogeneous Poisson process, superposition of a homogeneous Poisson process, determining the expected value of the number of events in an interval as a Poisson random variable approximation to a binomial random variable, the joint probability density function of the arrival times, and compound Poisson process were introduced. Some examples and problems including those related to the time of the earthquakes and the systems with different types of system failures were also presented.
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### Chapter 11. Renewal Process

Abstract
The first part of this chapter has been devoted to the basic terms related to a renewal process including the renewal function and renewal equation. The second part provided important theorems such as limit theorem, elementary renewal theorem, and renewal reward process, and some other properties such as the excess, age, and spread of a renewal process, inspection paradox, and central limit theorem for a renewal process. In the third part of the chapter, the alternating renewal process has been introduced as an example of a regenerative process. The concepts were illustrated by using some examples and problems including the ones related to the departures of the buses, and the system failures.
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### Chapter 12. An Introduction to Markov Chains

Abstract
This chapter was a compact introduction to both discrete-time and continuous-time Markov chains. The important concepts including the definition of discrete-time and continuous-time Markov chains, Chapman-Kolmogorov equations, reachability, communication, communication classes, recurrent and transient states, period of a state for a discrete-time Markov chain, the limiting probability of a state for a discrete-time and continuous-time Markov chain, and ergodic Markov chain were explained. Several examples and problems have been solved for the discrete-time Markov chains, and where relevant, state transition diagrams and tables have been used to facilitate the comprehension of the solutions.
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### Chapter 13. Special Discrete-Time Markov Chains

Abstract
This chapter has been devoted to some selected special discrete-time Markov chains with wide applications including random walk, simple random walk, simple symmetric random walk, Gambler’s ruin problem as a special simple random walk, branching process, hidden Markov chains, time-reversible discrete-time Markov chains, and Markov decision processes. Some illustrative examples and problems have been provided for each special discrete-time Markov chain.
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### Chapter 14. Continuous-Time Markov Chains

Abstract
The first part of this chapter presented the definition of a continuous-time Markov chain with two properties, and the introduction of a B&D process with some special examples such as homogeneous Poisson process as a pure birth process, and the population model as a B&D process. The second part of the chapter was an extension of a B&D process to the B&D queueing models including the general B&D queueing model, M/M/1 queueing model, and M/M/m queueing model. Finally, the last part of the chapter introduced Kolmogorov’s backward/forward equations, the infinitesimal generator matrix of a CTMC, and the time-reversibility condition for a CTMC. Several examples and problems including those related to the population model, the server systems, and the patient states have been provided and solved.
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### Chapter 15. An Introduction to Queueing Models

Abstract
This chapter was an introduction to the queueing models. The first part of the chapter introduced very basic concepts including Little’s law, Poisson Arrivals See Time Averages, and the Kendall’s notation for the classification of the queueing models. In the second part of the chapter, the balance equations for a general B&D queueing model, and the balance equations and Little’s law equations for an M/M/1 queueing model with infinite capacity and finite capacity have been provided. The adaptation of the basic formulations to the cases with more than one server, with varying arrival rates and departure rates, and with finite capacity has been illustrated with some examples and problems including the ones related to the arrivals of the customers, and the arrivals of the machines to a repair system.
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### Chapter 16. Introduction to Brownian Motion

Abstract
Brownian motion (BM) as a continuous-time extension to a simple symmetric random walk has been introduced in this chapter. The stationary and independent increments, normal distribution, and Markovian property have been provided as the properties of a standard Brownian motion. Brownian motion with drift, and geometric Brownian motion have also been defined as an extension to a standard Brownian motion. Some examples and problems have been provided to clarify the basic and other properties of a Brownian motion.
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### Chapter 17. Basics of Martingales

Abstract
This chapter was a brief introduction to the martingales. In the first part of the chapter; the definitions of a martingale, submartingale, supermartingale, and Doob type martingale were provided, and some examples were given to show how to verify whether a stochastic process is a martingale, submartingale or supermartingale. In the second part of the chapter, Azuma-Hoeffding inequality, Kolmogorov’s inequality for submartingales with an extension to martingales, and the martingale convergence theorem have been defined as the selected theorems of the martingales. Two problems were presented to show that a standard Brownian motion and a special type of Poisson process are martingales.
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### Chapter 18. Basics of Reliability Theory

Abstract
This chapter has been devoted to the reliability theory. In the first part of the chapter, some basic definitions for a nonrepairable item including the time to failure, availability, reliability function, failure rate function, mean time to failure, and mean residual life have been defined, and the concepts have been made clear with some illustrative examples. In the second part, some basic concepts for a repairable item including the mean up time, mean down time, mean time to repair, mean time between failures, and average availability have been presented with the basic definitions and some examples. In the third part, the basic concepts for the systems with independent components including the state vector, probability vector, structure function, and availability function have been defined for different structures with some examples. Finally, in the fourth part, the positive dependency and negative dependency have been defined for the systems with dependent components, and two methods, i.e. square-root method, and β-factor model have been introduced, and illustrated with some examples. The problems at the end of chapter were aimed to enhance the comprehension of very important basic concepts in reliability theory.
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### Backmatter

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