## Probability Concepts and Theory for EngineersA thorough introduction to the fundamentals of probability theoryThis 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

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 | |