Advanced mathematics for engineers with applications in stochastic processes

Topics in advanced mathematics for engineers, probability and statistics typically span three subject areas, are addressed in three separate textbooks and taught in three different courses in as many as three semesters. Due to this arrangement, students taking these courses have had to shelf some important and fundamental engineering courses until much later than is necessary. This practice has generally ignored some striking relations that exist between the seemingly separate areas of statistical concepts, such as moments and estimation of Poisson distribution parameters. On one hand, these concepts commonly appear in stochastic processes -- for instance, in measures on effectiveness in queuing models. On the other hand, they can also be viewed as applied probability in engineering disciplines -- mechanical, chemical, and electrical, as well as in engineering technology. There is obviously, an urgent need for a textbook that recognises the corresponding relationships between the various areas and a matching cohesive course that will see through to their fundamental engineering courses as early as possible. This book is designed to achieve just that. Its seven chapters, while retaining their individual integrity, flow from selected topics in advanced mathematics such as complex analysis and wavelets to probability, statistics and stochastic processes.

Author Notes

Aliakbar Montazer Haghigbi received Ms Ph.D. in Probability and Statistics from Case Western Reserve University, Cleveland, Ohio, USA, under the supervision of L. Takcs. He received his BA and MA in Mathematics from San Francisco State University, California. He is currently a Professor and Head of the Department of Mathematics at Prairie View AM University in Texas, USA. Previously he was employed by Institute of Statistics and Informatics (Iran), the Shaheed Beheshti University (Iran), and Benedict College (Columbia, SC) as a faculty, department chair, vice president for academic affairs and interim president. His research interests are probability, statistics, stochastic processes, and queueing theory. In addition to his extensive international publications, he has written/translated mathematics books in Farsi. His recent book Queueing Models in Industry and Business was published in 2008 by Nova Science Publishers, NY. Professor Haghighi is a life-member of American Mathematical Society (AMS) and Society for Industrial and Applied Mathematics (SIAM). He is the Co-founder and Editor-in-Chief of Applications and Applied Mathematics: An International Journal (AAM) that can be viewed at http://www.pvamu.edu/aam.
Jian-ao Lian received both his B.S. and M.S. degrees in mathematics from Xian Jiaotong University, Xian, China, in 1984 and 1987, respectively, and his Ph.D. degree in mathematics from Texas AM University, College Station, in 1993. He is currently a professor of mathematics at Prairie View AM University, Prairie View, TX, one of the nine campuses of the Texas AM University System in Texas. He is among the first to develop the orthonormal scaling functions and wavelets with symmetry by using the dilation factor three, as well as orthonormal scaling function vectors and multi-wavelets. Professor Lian is also a member of AMS and IEEE. His research interests include wavelets and applications, computer-aided geometric design, and signal and image processing.
Dimitar P. Mishev (Michev) obtained his M.S. and Ph.D. in mathematics from Sofia University, Sofia, Bulgaria. Currently, he is an associate professor in the Department of Mathematics at Prairie View AM University in Texas. Previously, he was associate professor and head of the Department of Differential Equations at Technical University, Sofia. Dr. Mishev has published many research papers, including three joint monographs, in his area of research interest-differential and difference equations and queueing theory, with Drumi D. Bainov and Aliakbar Montazer Haghighi.

Dedication	p. vii
Preface	p. xiii
Chapter 1 Introduction	p. 1
1.1 Functions of Several Variables	p. 1
1.2 Partial Derivatives, Gradient, & Divergence	p. 6
1.3 Functions of a Complex Variable	p. 17
1.4 Power Series & their Convergent Behavior	p. 20
1.5 Real-Valued Taylor Series & Maclaurin Series	p. 24
1.6 Power Series Representation of Analytic Functions	p. 25
1.6.1 Derivative and Analytic Functions	p. 25
1.6.2 Line Integral in the Complex Plane	p. 29
1.6.3 Cauchy's Integral Theorem for Simply Connected Domains	p. 30
1.6.4 Cauchy's Integral Theorem for Multiply Connected Domains	p. 32
1.6.5 Cauchy's Integral Formula	p. 33
1.6.6 Cauchy's Integral Formula for Derivatives	p. 34
1.6.7 Taylor and Maclaurin Series of Complex-Valued Functions	p. 35
1.6.8 Taylor Polynomials and their Applications	p. 37
Chapter 1 Exercises	p. 37
Chapter 2 Fourier and Wavelet Analysis	p. 47
2.1 Vector Spaces and Orthogonality	p. 47
2.2 Fourier Series & its Convergent Behavior	p. 55
2.3 Fourier Cosine and Sine Series & Half-Range Expansions	p. 58
2.4 Fourier Series and Partial Differential Equations	p. 60
2.5 Fourier Transform & Inverse Fourier Transform	p. 63
2.6 Properties of Fourier Transform & Convolution Theorem	p. 63
2.7 Discrete Fourier Transform & Fast Fourier Transform	p. 64
2.8 Classical Haar Scaling Function & Haar Wavelets	p. 68
2.9 Daubechies Orthonormal Scaling Functions & Wavelets	p. 70
2.10 Multiresolution Analysis in General	p. 80
2.11 Wavelet Transform & Inverse Wavelet Transform	p. 81
2.12 Other Wavelets	p. 82
2.12.1 Compactly Supported Spline Wavelets	p. 82
2.12.2 Morlet Wavelets	p. 85
2.12.3 Gaussian Wavelets	p. 87
2.12.4 Biorthogonal Wavelets	p. 88
2.12.5 CDF 5/3 Wavelets	p. 90
2.12.6 CDF 9/7 Wavelets	p. 90
Chapter 2 Exercises	p. 95
Chapter 3 Laplace Transform	p. 101
3.1 Definitions of Laplace Transform & Inverse Laplace Transform	p. 101
3.2 First Shifting Theorem	p. 104
3.3 Laplace Transform of Derivatives	p. 105
3.4 Solving Initial-Value Problems by Laplace Transform	p. 106
3.5 Heaviside Function & Second Shifting Theorem	p. 107
3.6 Solving Initial-Value Problems with Discontinuous Inputs	p. 109
3.7 Short Impulse & Dirac's Delta Functions	p. 110
3.8 Solving Initial-Value Problems with Impulse Inputs	p. 111
3.9 Application of Laplace Transform to Electric Circuits	p. 112
3.10 Table of Laplace Transforms	p. 113
Chapter 3 Exercises	p. 114
Chapter 4 Probability	p. 121
4.1 Introduction	p. 121
4.2 Counting Techniques	p. 128
4.3 Tree Diagrams	p. 135
4.4 Conditional probability and Independence	p. 136
4.5 The Law of Total probability	p. 141
4.6 Discrete Random Variables	p. 151
4.7 Discrete Probability Distributions	p. 154
4.8 Random Vectors	p. 171
4.9 Conditional Distribution and Independence	p. 177
4.10 Discrete Moments	p. 181
4.11 Continuous Random Variables and Distributions	p. 193
4.12 Continuous Random Vector	p. 219
4.13 Functions of a Random Variable	p. 222
Chapter 4 Exercises	p. 229
Chapter 5 Statistics	p. 249
Part 1: Descriptive Statistics	p. 251
5.1 Basics Statistical Concepts	p. 251
5.1.1 Measures of Central Tendency	p. 253
5.1.2 Organization of Data	p. 256
5.1.3 Measures of Variability	p. 267
Part 2: Inferential Statistics	p. 270
5.2 Estimation	p. 270
5.2.1 Point Estimation	p. 270
5.2.1.a Method of Moments	p. 274
5.2.1.b Maximum Likelihood Estimator (MLE)	p. 275
5.2.2 Interval Estimation	p. 280
5.3 Hypothesis Testing	p. 286
5.4 Inference on Two Small-Sample Means	p. 294
5.4 Case 1 Population Variances Not Necessarily Equal, But Unknown (Confidence Interval)	p. 294
5.4 Case 2 Population Variances Equal, But Unknown (Confidence Interval)	p. 297
5.4 Case 3 Population Variances Equal, But Unknown (Testing Null Hypothesis)	p. 299
5.5 Analysis of Variance (ANOVA)	p. 300
5.6 Linear Regression	p. 305
Part 3: Applications of Statistics	p. 317
5.7 Reliability	p. 317
5.8 Estimation of Reliability	p. 327
5.8.1 Estimation of the Reliability by (MLE) Method	p. 329
5.8.2 Estimation of Reliability by Shrinkage Procedures	p. 331
5.8.2.a Shrinking Towards a Pre-specified R	p. 331
5.8.2.b Shrinking using the p-value of the LRT	p. 333
5.8.3 Estimation of Reliability by the Method of Moments	p. 336
5.9 Reliability Computation using Logistic Distributions	p. 341
5.10 Reliability Computation using Extreme-Value Distributions	p. 348
5.11 Simulation for computing Reliability using Logistic and Extreme-Value Distributions	p. 353
5.12 Basics Concepts About Up-and-Down Design	p. 362
5.13 Up-and-Down Design	p. 368
Chapter 5 Exercises	p. 371
Chapter 6 Differential and Difference Equations	p. 387
6.1 Introduction	p. 387
6.2 Linear First Order Difference Equations	p. 392
6.3 Behavior of Solutions of the First Order Difference Equations	p. 396
Generating Functions and their Applications in Solution of Difference Equations	p. 405
6.5 Differential Difference Equations	p. 422
Chapter 6 Exercises	p. 424
Chapter 7 Stochastic Processes and Their Applications	p. 427
7.1 Introduction	p. 427
7.2 Markov Chain and Markov Process	p. 432
7.3 Classification of States of a Markov Chain	p. 442
7.4 Random Walk Process	p. 445
7.5 Birth-Death (B-D) Processes	p. 453
7.6 Pure Birth process - Poisson Process	p. 456
7.7 Introduction of Queueing Process and its Historical Development	p. 471
7.8 Stationary Distributions	p. 479
7.9 Waiting Time	p. 487
7.10 Single-server Queue with Feedback	p. 490
7.10.1 M/M/1/N with Feedback	p. 491
7.10.2 Infinite Single-Server Queue with Feedback	p. 492
7.10.3 The Stationary Distribution of the Feedback Queue Size	p. 497
7.10.4 The Stationary Distribution of Sojourn Time of the nth Customer	p. 500
7.10.5 The Moments of Sojourn Time of the nth Customer	p. 503
Chapter 7 Exercises	p. 507
Appendices (Distribution Tables)	p. 515
References	p. 535
Answers to Selected Exercises	p. 543
Index	p. 547

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