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Probability and Stochastic Processes With a View Toward Applications Contents 1 The Basic Probability Model 1 Introduction 1 Fair Coin-tossing 1 Models with a Finite Number of Outcomes 2 The Heuristics of Model-building 3 The Additivity of Probability 4 Infinite Outcomes Spaces 7 Probabilities as Integrals 14 Summary 18 2 Some Classical Models 19 Introduction 19 Equally Likely Outcomes and Independent Trials 19 The Binomial Distribution 22 The Hypergeometric Distribution 27 The Multinomial Distribution 30 The Poisson Distribution 32 The Exponential Distribution 37 The Uniform Distribution 40 Summary 42 3 Random Variables 44 The Definitions 44 Remarks on Random Variables 49 The Three Types of Distributions of Random Variables 50 The Cumulative Distribution Function 53 Densities of Functions of Random Variables 55 The Expected Value of a Random Variable 56 Expectation of a Function of a Random Variable 61 The Joint Distribution of Two Random Variables 65 Expectations of Functions of Two Random Variables 70 Covariance and the Correlation Coefficient 74 Many Random Variables 76 An Infinite Number of Random Variables 79 Summary 82 4 Independent Random Variables 84 Definitions 84 Independence is a Family Property 89 More on the Poisson Process 93 Independence, Expectations, and Variances 98 Identically Distributed Random Variables, and the Law of Averages 102 Sums of Small Independent Components are Normally Distributed 106 How Deviant is a Deviation? 112 How Good is the Normal Approximation? 117 The Binomial Distribution 119 The Poisson Distribution 120 The Exponential Distribution 122 The Uniform Distribution 123 Summary 124 5 Conditional Probability 126 Introduction 126 Some Examples 130 The Distribution of X Given Y = y 134 A Useful Rule 137 The Addition Rule for Conditional Probabilities 138 Conditional Expectation 143 Dependent Sequences of Random Variables 147 Summary 151 6 Markov Chains 152 Definitions 152 Some Examples 155 Some General Properties of Markov Motion 164 The Stability of a Markov System 168 Integer States and Recurrence Times 175 The Stability Problem Solved for Integer States 180 The Difference Equation Method 188 Summary 194 7 Continuous Time Markov Processes 196 Models for Continuous Time Processes 196 Continuous Time Integer-valued Markov Processes 197 The Infinitesimal Transition Scheme 200 The Differential Equations for the Transition Probabilities 205 The Steady-state Distributions 208 How Does a Markov Process Operate? 211 The Difference Equation 215 Summary 216 8 Vector Independence and the Multivariate Normal Distribution 217 Introduction 217 The Covariance Matrix and Means Vector 218 Independence of Random Vectors 221 Sums of Vector Variables 224 The Central Limit Theorem 226 The Multivariate Normal Distribution 233 Properties of the Multivariate Normal Distribution 242 Gaussian Processes 246 Summary 247 9 Stationary Time Series 249 Introduction 249 Gaussian and Second-order Stationary Processes 253 Almost Periodic Processes 257 The General Frequency Representation 264 Linear Systems 274 White Noise and the Gauss-Markov Process 281 The General Prediction Problem 292 Linear Prediction and Filtering 300 Modeling and Ergodicity 313 Summary 318 Table 320 Index 321