## Probability for ApplicationsObjecti'ves. As the title suggests, this book provides an introduction to probability designed to prepare the reader for intelligent and resourceful applications in a variety of fields. Its goal is to provide a careful exposition of those concepts, interpretations, and analytical techniques needed for the study of such topics as statistics, introductory random processes, statis tical communications and control, operations research, or various topics in the behavioral and social sciences. Also, the treatment should provide a background for more advanced study of mathematical probability or math ematical statistics. The level of preparation assumed is indicated by the fact that the book grew out of a first course in probability, taken at the junior or senior level by students in a variety of fields-mathematical sciences, engineer ing, physics, statistics, operations research, computer science, economics, and various other areas of the social and behavioral sciences. Students are expected to have a working knowledge of single-variable calculus, including some acquaintance with power series. Generally, they are expected to have the experience and mathematical maturity to enable them to learn new concepts and to follow and to carry out sound mathematical arguments. While some experience with multiple integrals is helpful, the essential ideas can be introduced or reviewed rather quickly at points where needed. |

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### Contents

3 | |

Probability Systems | 25 |

2a The Sigma Algebra of Events | 44 |

Independence of Events | 73 |

Conditional Independence of Events 89 | 88 |

Composite Trials | 123 |

Random Variables and Probabilities | 145 |

7a Borel Sets Random Variables and Borel Functions | 160 |

Variance and Standard Deviation | 355 |

Covariance Correlation and Linear Regression | 371 |

Convergence in Probability Theory 393 | 392 |

Transform Methods | 409 |

Conditional Expectation | 443 |

19a Some Theoretical Details | 481 |

Random Selection and Counting Processes | 491 |

Poisson Processes | 541 |

Random Vectors and Joint Distributions | 197 |

Independence of Random Vectors 215 | 214 |

Functions of Random Variables | 233 |

11a Some Properties of the Quantile Function | 266 |

Mathematical Expectation | 273 |

Expectation and Integrals | 287 |

13a Supplementary Theoretical Details | 312 |

Properties of Expectation | 323 |

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### Common terms and phrases

arrival assigned basic space Bernoulli sequence Bernoulli trials binomial Boolean combination Borel function Borel sets Chapter class of events codomain component trial conditional expectation conditional independence conditional probability consider convergence countable counting process defined Definition distribution function E[IM equivalent event determined Example exponential expression failure Figure finite Fx(t geometric given Hence implies increments indicator functions inequality interarrival interval joint density function joint distribution Lebesgue integral lemma linearity mathematical expectation matrix mean value measurable function minterm moment-generating function nonnegative normal distribution obtain outcome P(AB pair parameter Poisson process population probability distribution probability mass probability measure problem product rule Proof properties quantile function random quantity random vector Remark result sample Section selected sigma algebra simple random variable Solution statistics subchain subset Suppose Theorem theory tion transition unit Var[X variance zero