## Applied Probability and StatisticsThis book is based mainly on the lecture notes that I have been using since 1993 for a course on applied probability for engineers that I teach at the Ecole Polytechnique de Montreal. This course is given to electrical, computer and physics engineering students, and is normally taken during the second or third year of their curriculum. Therefore, we assume that the reader has acquired a basic knowledge of differential and integral calculus. The main objective of this textbook is to provide a reference that covers the topics that every student in pure or applied sciences, such as physics, computer science, engineering, etc., should learn in probability theory, in addition to the basic notions of stochastic processes and statistics. It is not easy to find a single work on all these topics that is both succinct and also accessible to non-mathematicians. Because the students, who for the most part have never taken a course on prob ability theory, must do a lot of exercises in order to master the material presented, I included a very large number of problems in the book, some of which are solved in detail. Most of the exercises proposed after each chapter are problems written es pecially for examinations over the years. They are not, in general, routine problems, like the ones found in numerous textbooks. |

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### Contents

Introduction1 | 2 |

12 Examples of Applications | 3 |

13 Relative Frequencies | 5 |

Elementary Probabilities | 7 |

22 Probability | 10 |

23 Combinatorial Analysis | 13 |

24 Conditional Probability | 18 |

25 Independence | 21 |

412 Exercises Problems and Multiple Choice Questions | 195 |

Stochastic Processes | 220 |

52 Characteristics of Stochastic Processes | 222 |

53 Markov Chains | 225 |

54 The Poisson Process | 228 |

55 The Wiener Process | 232 |

56 Stationarity | 235 |

57 Ergodicity | 238 |

26 Exercises Problems and Multiple Choice Questions Solved Exercises | 26 |

Random Variables | 55 |

32 The Distribution Function | 57 |

33 The Probability Mass and Density Functions | 64 |

34 Important Discrete Random Variables | 70 |

35 Important Continuous Random Variables | 82 |

36 Transformations | 92 |

37 Mathematical Expectation and Variance | 95 |

38 Transforms | 103 |

39 Reliability | 108 |

310 Exercises Problems and Multiple Choice Questions Solved Exercises | 111 |

Random Vectors | 157 |

42 Random Vectors of Dimension 2 | 158 |

43 Conditionals | 166 |

44 Random Vectors of Dimension n 2 | 170 |

45 Transformations of Random Vectors | 172 |

46 Covariance and Correlation | 176 |

47 Multinormal Distribution | 179 |

48 Estimation of a Random Variable | 182 |

49 Linear Combinations | 185 |

410 The Laws of Large Numbers | 188 |

411 The Central Limit Theorem | 189 |

58 Exercises Problems and Multiple Choice Questions Solved Exercises | 240 |

Estimation and Testing | 253 |

62 Estimation by Confidence Intervals | 258 |

63 Pearsons ChiSquare GoodnessofFit Test | 262 |

64 Tests of Hypotheses on the Parameters | 266 |

65 Exercises Problems and Multiple Choice Questions Supplementary Exercises | 279 |

Simple Linear Regression | 307 |

72 Tests of Hypotheses | 310 |

73 Confidence Intervals and Ellipses | 313 |

74 The Coefficient of Determination | 315 |

76 Curvilinear Regression | 318 |

77 Correlation | 321 |

78 Exercises Problems and Multiple Choice Questions Supplementary Exercises | 324 |

Mathematical Formulas | 339 |

Quantiles of the Sampling Distributions | 341 |

Classification of the Exercises | 344 |

Answers to the Multiple Choice Questions | 347 |

Answers to Selected Supplementary Exercises | 349 |

351 | |

353 | |

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