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Introduction | |

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The Analysis of Random Experiments | |

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Probability in Electrical and Computer Engineering | |

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Signal detection and classification | |

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Speech modeling and recognition | |

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Coding and data transmission | |

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Computer networks | |

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Outline of the Book | |

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The Probability Model | |

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The Algebra of Events | |

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Basic operations | |

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Representation of the sample space | |

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Probability of Events | |

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Defining probability | |

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Statistical independence | |

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Some Applications | |

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Repeated independent trials | |

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Problems involving counting | |

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Network reliability | |

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Conditional Probability and Bayes' Rule | |

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Conditional probability | |

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Event trees | |

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Bayes' rule | |

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More Applications | |

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The binary communication channel | |

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Measuring information and coding | |

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Summary | |

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Random Variables and Transformations | |

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Discrete Random Variables | |

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Common Discrete Probability Distributions | |

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Bernoulli random variable | |

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Binomial random variable | |

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Geometric random variable | |

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Poisson random variable | |

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Discrete uniform random variable | |

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Continuous Random Variables | |

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Probabilistic description | |

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More about the PDF | |

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A relation to discrete random variables | |

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Solving problems | |

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Common Continuous Probability Density Functions | |

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Uniform random variable | |

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Exponential random variable | |

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Gaussian random variable | |

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CDF and PDF for Discrete and Mixed Random Variables | |

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Discrete random variables | |

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Mixed random variables | |

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Fransformation of Random Variables | |

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When the transformation is invertible | |

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When the transformation is not invertible | |

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When the transformation has discontinuities or flat regions | |

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Instributions Conditioned on an Event | |

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Applications | |

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Optimal signal detection | |

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Object classification | |

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Summary | |

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Expectation, Moments, and Generating Functions | |

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Expectation of a Random Variable | |

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Discrete random variable | |

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Continuous random variable | |

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Invariance of expectation | |

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Properties of expectation | |

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Expectation conditioned on an event | |

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Meanents of a Distribution | |

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Central moments | |

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Properties of variance | |

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Some higher-order moments | |

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Generating Functions | |

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The moment generating function | |

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The probability generating function | |

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Application: Entropy and Source Coding | |

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Summary | |

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Two and More Random Variables | |

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Two Discrete Random Variables | |

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The joint PMF | |

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Independent random variables | |

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Conditional PMFs for discrete random variables | |

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Bayes' rule for discrete random variables | |

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Two Continuous Random Variables | |

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Joint distributions | |

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Marginal PDFs: Projections of the joint density | |

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Conditional PDFs: Slices of the joint density | |

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Bayes' rule for continuous random variables | |

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Expectation and Correlation | |

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Correlation and covariance | |

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Conditional expectation | |

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Gaussian Random Variables | |

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Multiple Random Variables | |

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PDFs for multiple random variables | |

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Sums of random variables | |

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Sums of Some Common Random Variables | |

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Bernoulli random variables | |

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Geometric random variables | |

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Exponential random variables | |

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Gaussian random variables | |

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Squared Gaussian random variables | |

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Random Vectors | |

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Cumulative distribution and density functions | |

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Expectation and moments | |

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Multivariate Gaussian density function | |

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Transformations of random vectors | |

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An Application to Signal Detection | |

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Summary | |

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Inequalities, Limit Theorems, and Parameter Estimation | |

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Inequalities | |

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Markov inequality | |

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Chebyshev inequality | |

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One-sided Chebyshev inequality | |

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Other inequalities | |

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Convergence and Limit Theorems | |

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Laws of large numbers | |

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Central limit theorem | |

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Estimation of Parameters | |

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Estimates and properties | |

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Sample mean and variance | |

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Maximum Likelihood Estimation | |

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Application to Signal Estimation | |

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Summary | |

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Random Processes | |

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Random Process | |

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The ensemble | |

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First and Second Moments of a Random Process | |

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Mean | |

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Autocorrelation and autocovariance functions | |

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Cross-correlation function | |

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Properties: Independence, Stationarity, and Ergodicity | |

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Statistical independence | |

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Strict sense stationarity | |

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Wide sense stationarity | |

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Properties of correlation functions | |

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Time averages and ergodic random processes | |

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Power Spectral Density | |

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Properties of the power spectral density | |

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Cross-power spectral density | |

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White noise | |

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An application | |

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Noise Sources | |

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Thermal noise | |

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Quantization noise | |

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Response of Linear Systems | |

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Linear time-invariant system | |

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Output mean | |

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Cross-correlation functions and cross-power spectra | |

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Autocorrelation function and power spectral density of system output | |

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Response of linear systems: discrete case | |

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Summary | |

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Markov and Poisson Random Processes | |

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The Poisson Model | |

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Derivation of the Poisson model | |

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The Poisson process | |

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An application of the Poisson process: the random telegraph signal | |

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Additional remarks about the Poisson process | |

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Discrete-Time Markov Chains | |

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Definitions and dynamic equations | |

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Higher-order transition probabilities | |

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Limiting state probabilities | |

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Continuous-Time Markov Chains | |

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Simple server system | |

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Analysis of continuous-time Markov chains | |

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Special condition for birth and death processes | |

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Basic Queueing Theory | |

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The single-server system | |

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Little's formula | |

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The single-server system with finite capacity | |

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The multi-server system | |

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Summary | |

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Basic Combinatorics | |

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The Rule of Product | |

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Permutations | |

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Combinations | |

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The Unit Impulse | |

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The Impulse | |

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Properties | |

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Index | |