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Preface | |
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Acknowledgment | |
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Basic Probability Concepts | |
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Introduction | |
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Sample Space and Events | |
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Definitions of Probability | |
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Axiomatic Definition | |
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Relative-Frequency Definition | |
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Classical Definition | |
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Applications of Probability | |
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Reliability Engineering | |
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Quality Control | |
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Channel Noise | |
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System Simulation | |
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Elementary Set Theory | |
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Set Operations | |
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Number of Subsets of a Set | |
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Venn Diagram | |
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Set Identities | |
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Duality Principle | |
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Properties of Probability | |
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Conditional Probability | |
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Total Probability and the Bayes' Theorem | |
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Tree Diagram | |
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Independent Events | |
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Combined Experiments | |
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Basic Combinatorial Analysis | |
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Permutations | |
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Circular Arrangement | |
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Applications of Permutations in Probability | |
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Combinations | |
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The Binomial Theorem | |
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Stirling's Formula | |
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Applications of Combinations in Probability | |
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Reliability Applications | |
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Chapter Summary | |
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Problems | |
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References | |
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Random Variables | |
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Introduction | |
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Definition of a Random Variable | |
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Events Defined by Random Variables | |
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Distribution Functions | |
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Discrete Random Variables | |
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Obtaining the PMF from the CDF | |
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Continuous Random Variables | |
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Chapter Summary | |
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Problems | |
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Moments of Random Variables | |
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Introduction | |
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Expectation | |
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Expectation of Nonnegative Random Variables | |
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Moments of Random Variables and the Variance | |
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Conditional Expectations | |
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The Chebyshev Inequality | |
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The Markov Inequality | |
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Chapter Summary | |
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Problems | |
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Special Probability Distributions | |
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Introduction | |
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The Bernoulli Trial and Bernoulli Distribution | |
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Binomial Distribution | |
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Geometric Distribution | |
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Modified Geometric Distribution | |
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"Forgetfulness" Property of the Geometric Distribution | |
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Pascal (or Negative Binomial) Distribution | |
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Hypergeometric Distribution | |
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Poisson Distribution | |
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Poisson Approximation to the Binomial Distribution | |
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Exponential Distribution | |
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"Forgetfulness" Property of the Exponential Distribution | |
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Relationship between the Exponential and Poisson Distributions | |
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Erlang Distribution | |
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Uniform Distribution | |
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The Discrete Uniform Distribution | |
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Normal Distribution | |
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Normal Approximation to the Binomial Distribution | |
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The Error Function | |
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The Q-Function | |
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The Hazard Function | |
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Chapter Summary | |
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Problems | |
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Multiple Random Variables | |
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Introduction | |
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Joint CDFs of Bivariate Random Variables | |
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Properties of the Joint CDF | |
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Discrete Random Variables | |
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Continuous Random Variables | |
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Determining Probabilities from a Joint CDF | |
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Conditional Distributions | |
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Conditional PMF for Discrete Random Variables | |
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Conditional PDF for Continuous Random Variables | |
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Conditional Means and Variances | |
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Simple Rule for Independence | |
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Covariance and Correlation Coefficient | |
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Many Random Variables | |
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Multinomial Distributions | |
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Chapter Summary | |
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Problems | |
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Functions of Random Variables | |
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Introduction | |
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Functions of One Random Variable | |
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Linear Functions | |
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Power Functions | |
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Expectation of a Function of One Random Variable | |
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Moments of a Linear Function | |
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Sums of Independent Random Variables | |
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Moments of the Sum of Random Variables | |
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Sum of Discrete Random Variables | |
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Sum of Independent Binomial Random Variables | |
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Sum of Independent Poisson Random Variables | |
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The Spare Parts Problem | |
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Minimum of Two Independent Random Variables | |
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Maximum of Two Independent Random Variables | |
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Comparison of the Interconnection Models | |
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Two Functions of Two Random Variables | |
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Application of the Transformation Method | |
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Laws of Large Numbers | |
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The Central Limit Theorem | |
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Order Statistics | |
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Chapter Summary | |
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Problems | |
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Transform Methods | |
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Introduction | |
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The Characteristic Function | |
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Moment-Generating Property of the Characteristic Function | |
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The s-Transform | |
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Moment-Generating Property of the s-Transform | |
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The s-Transforms of Some Well-Known PDFs | |
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The s-Transform of the PDF of the Sum of Independent Random Variables | |
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The z-Transform | |
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Moment-Generating Property of the z-Transform | |
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The z-Transform of the Bernoulli Distribution | |
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The z-Transform of the Binomial Distribution | |
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The z-Transform of the Geometric Distribution | |
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The z-Transform of the Poisson Distribution | |
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The z-Transform of the PMF of the Sum of Independent Random Variables | |
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The z-Transform of the Pascal Distribution | |
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Random Sum of Random Variables | |
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Chapter Summary | |
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Problems | |
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Introduction to Random Processes | |
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Introduction | |
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Classification of Random Processes | |
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Characterizing a Random Process | |
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Mean and Autocorrelation Function of a Random Process | |
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The Autocovariance Function of a Random Process | |
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Crosscorrelation and Crosscovariance Functions | |
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Review of Some Trigonometric Identities | |
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Stationary Random Processes | |
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Strict-Sense Stationary Processes | |
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Wide-Sense Stationary Processes | |
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Ergodic Random Processes | |
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Power Spectral Density | |
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White Noise | |
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Discrete-Time Random Processes | |
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Mean, Autocorrelation Function, and Autocovariance Function | |
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Power Spectral Density | |
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Sampling of Continuous-Time Processes | |
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Chapter Summary | |
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Problems | |
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Linear Systems with Random Inputs | |
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Introduction | |
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Overview of Linear Systems with Deterministic Inputs | |
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Linear Systems with Continuous-Time Random Inputs | |
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Linear Systems with Discrete-Time Random Inputs | |
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Autoregressive Moving Average Process | |
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Moving Average Process | |
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Autoregressive Process | |
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ARMA Process | |
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Chapter Summary | |
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Problems | |
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Some Models of Random Processes | |
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Introduction | |
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The Bernoulli Process | |
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Random Walk | |
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Gambler's Ruin | |
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The Gaussian Process | |
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White Gaussian Noise Process | |
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Poisson Process | |
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Counting Processes | |
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Independent Increment Processes | |
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Stationary Increments | |
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Definitions of a Poisson Process | |
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Interarrival Times for the Poisson Process | |
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Conditional and Joint PMFs for Poisson Processes | |
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Compound Poisson Process | |
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Combinations of Independent Poisson Processes | |
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Competing Independent Poisson Processes | |
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Subdivision of a Poisson Process and the Filtered Poisson Process | |
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Random Incidence | |
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Nonhomogeneous Poisson Process | |
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Markov Processes | |
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Discrete-Time Markov Chains | |
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State Transition Probability Matrix | |
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The n-Step State Transition Probability | |
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State Transition Diagrams | |
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Classification of States | |
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Limiting-State Probabilities | |
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Doubly Stochastic Matrix | |
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Continuous-Time Markov Chains | |
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Birth and Death Processes | |
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Gambler's Ruin as a Markov Chain | |
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Chapter Summary | |
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Problems | |
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Introduction to Statistics | |
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Introduction | |
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Sampling Theory | |
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The Sample Mean | |
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The Sample Variance | |
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Sampling Distributions | |
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Estimation Theory | |
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Point Estimate, Interval Estimate, and Confidence Interval | |
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Maximum Likelihood Estimation | |
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Minimum Mean Squared Error Estimation | |
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Hypothesis Testing | |
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Hypothesis Test Procedure | |
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Type I and Type II Errors | |
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One-Tailed and Two-Tailed Tests | |
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Curve Fitting and Linear Regression | |
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Chapter Summary | |
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Problems | |
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Table for the CDF of the Standard Normal Random Variable | |
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Bibliography | |
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Index | |