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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 | |