Probability Concepts and Theory for Engineers

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John Wiley & Sons, Feb 18, 2011 - Technology & Engineering - 622 pages
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A thorough introduction to the fundamentals of probability theory

This book offers a detailed explanation of the basic models and mathematical principles used in applying probability theory to practical problems. It gives the reader a solid foundation for formulating and solving many kinds of probability problems for deriving additional results that may be needed in order to address more challenging questions, as well as for proceeding with the study of a wide variety of more advanced topics.

Great care is devoted to a clear and detailed development of the ‘conceptual model' which serves as the bridge between any real-world situation and its analysis by means of the mathematics of probability. Throughout the book, this conceptual model is not lost sight of. Random variables in one and several dimensions are treated in detail, including singular random variables, transformations, characteristic functions, and sequences. Also included are special topics not covered in many probability texts, such as fuzziness, entropy, spherically symmetric random variables, and copulas.

Some special features of the book are:

  • a unique step-by-step presentation organized into 86 topical Sections, which are grouped into six Parts
  • over 200 diagrams augment and illustrate the text, which help speed the reader's comprehension of the material
  • short answer review questions following each Section, with an answer table provided, strengthen the reader's detailed grasp of the material contained in the Section
  • problems associated with each Section provide practice in applying the principles discussed, and in some cases extend the scope of that material
  • an online separate solutions manual is available for course tutors.

The various features of this textbook make it possible for engineering students to become well versed in the ‘machinery' of probability theory. They also make the book a useful resource for self-study by practicing engineers and researchers who need a more thorough grasp of particular topics.

 

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Contents

The Probabilistic Experiment
Elements and Sets
Elementary Set Operations
Functions
Multiple and Infinite Set Operations
Additive Set Functions
The Probability Function
Simple Probability Arithmetic
The Sum of Two Discrete Random Variables
nDimensional Random Variables
Absolutely Continuous nDimensional R V
Rotations and the Bivariate Gaussian Distribution
Several Statistically Independent Random Variables
Singular Distributions in One Dimension
Conditional Induced Distribution Given an Event
Resolving a Distribution into Components of Pure Type

The Approach to Elementary Probability
About Probability Problems
Equally Likely Possible Outcomes
Conditional Probability
Conditional Probability Distributions
Independent Events
Classes of Independent Events
Possible Outcomes Represented as Ordered kTuples
Product Experiments and Product Spaces
Product Probability Spaces
Dependence Between the Components in an Ordered k Tuple
Multiple Observations Without Regard to Order
Unordered Sampling with Replacement
More Complicated Discrete Probability Problems
Uncertainty and Randomness
Fuzziness
Summary
Introduction
The Binomial Distribution
General Definition of a Random Variable
The Probability Density Function
The Gaussian Distribution
Two Arbitrary Random Variables
TwoDimensional Distribution Functions
TwoDimensional Density Functions
Two Statistically Independent Random Variables
Two Statistically Independent Random
Transformations and Multiple Random
Transformation of a TwoDimensional Random
Conditional Distribution Given the Value of a Random
Parameters for Describing Random
Higher Moments
Expectation of a Function of a Random Variable
The Variance of a Function of a Random Variable
Test Sampling
Conditional Expectation with Respect to an Event
Covariance and Correlation Coefficient
The Correlation Coefficient as Parameter in a Joint
More General Kinds of Dependence Between Random
The Covariance Matrix
Random Variables as the Elements of a Vector Space
Estimation
The Stieltjes Integral
Further Topics in Random Variables
The Characteristic Function
Characteristic Function of a Transformed Random
Characteristic Function of a Multidimensional
Several Jointly Gaussian Random Variables
Spherically Symmetric Vector Random Variables
Entropy Associated with Random Variables
Copulas
Sequences of Random Variables
Convergent Sequences and Laws of Large Numbers
Convergence of Probability Distributions and
Appendices
Notation and Abbreviations
Symbols and markings
Copyright

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