| |

| |

Preface | |

| |

| |

Acknowledgment | |

| |

| |

| |

Basic Probability Concepts | |

| |

| |

| |

Introduction | |

| |

| |

| |

Sample Space and Events | |

| |

| |

| |

Definitions of Probability | |

| |

| |

| |

Axiomatic Definition | |

| |

| |

| |

Relative-Frequency Definition | |

| |

| |

| |

Classical Definition | |

| |

| |

| |

Applications of Probability | |

| |

| |

| |

Reliability Engineering | |

| |

| |

| |

Quality Control | |

| |

| |

| |

Channel Noise | |

| |

| |

| |

System Simulation | |

| |

| |

| |

Elementary Set Theory | |

| |

| |

| |

Set Operations | |

| |

| |

| |

Number of Subsets of a Set | |

| |

| |

| |

Venn Diagram | |

| |

| |

| |

Set Identities | |

| |

| |

| |

Duality Principle | |

| |

| |

| |

Properties of Probability | |

| |

| |

| |

Conditional Probability | |

| |

| |

| |

Total Probability and the Bayes' Theorem | |

| |

| |

| |

Tree Diagram | |

| |

| |

| |

Independent Events | |

| |

| |

| |

Combined Experiments | |

| |

| |

| |

Basic Combinatorial Analysis | |

| |

| |

| |

Permutations | |

| |

| |

| |

Circular Arrangement | |

| |

| |

| |

Applications of Permutations in Probability | |

| |

| |

| |

Combinations | |

| |

| |

| |

The Binomial Theorem | |

| |

| |

| |

Stirling's Formula | |

| |

| |

| |

Applications of Combinations in Probability | |

| |

| |

| |

Reliability Applications | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Problems | |

| |

| |

| |

References | |

| |

| |

| |

Random Variables | |

| |

| |

| |

Introduction | |

| |

| |

| |

Definition of a Random Variable | |

| |

| |

| |

Events Defined by Random Variables | |

| |

| |

| |

Distribution Functions | |

| |

| |

| |

Discrete Random Variables | |

| |

| |

| |

Obtaining the PMF from the CDF | |

| |

| |

| |

Continuous Random Variables | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Problems | |

| |

| |

| |

Moments of Random Variables | |

| |

| |

| |

Introduction | |

| |

| |

| |

Expectation | |

| |

| |

| |

Expectation of Nonnegative Random Variables | |

| |

| |

| |

Moments of Random Variables and the Variance | |

| |

| |

| |

Conditional Expectations | |

| |

| |

| |

The Chebyshev Inequality | |

| |

| |

| |

The Markov Inequality | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Problems | |

| |

| |

| |

Special Probability Distributions | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Bernoulli Trial and Bernoulli Distribution | |

| |

| |

| |

Binomial Distribution | |

| |

| |

| |

Geometric Distribution | |

| |

| |

| |

Modified Geometric Distribution | |

| |

| |

| |

"Forgetfulness" Property of the Geometric Distribution | |

| |

| |

| |

Pascal (or Negative Binomial) Distribution | |

| |

| |

| |

Hypergeometric Distribution | |

| |

| |

| |

Poisson Distribution | |

| |

| |

| |

Poisson Approximation to the Binomial Distribution | |

| |

| |

| |

Exponential Distribution | |

| |

| |

| |

"Forgetfulness" Property of the Exponential Distribution | |

| |

| |

| |

Relationship between the Exponential and Poisson Distributions | |

| |

| |

| |

Erlang Distribution | |

| |

| |

| |

Uniform Distribution | |

| |

| |

| |

The Discrete Uniform Distribution | |

| |

| |

| |

Normal Distribution | |

| |

| |

| |

Normal Approximation to the Binomial Distribution | |

| |

| |

| |

The Error Function | |

| |

| |

| |

The Q-Function | |

| |

| |

| |

The Hazard Function | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Problems | |

| |

| |

| |

Multiple Random Variables | |

| |

| |

| |

Introduction | |

| |

| |

| |

Joint CDFs of Bivariate Random Variables | |

| |

| |

| |

Properties of the Joint CDF | |

| |

| |

| |

Discrete Random Variables | |

| |

| |

| |

Continuous Random Variables | |

| |

| |

| |

Determining Probabilities from a Joint CDF | |

| |

| |

| |

Conditional Distributions | |

| |

| |

| |

Conditional PMF for Discrete Random Variables | |

| |

| |

| |

Conditional PDF for Continuous Random Variables | |

| |

| |

| |

Conditional Means and Variances | |

| |

| |

| |

Simple Rule for Independence | |

| |

| |

| |

Covariance and Correlation Coefficient | |

| |

| |

| |

Many Random Variables | |

| |

| |

| |

Multinomial Distributions | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Problems | |

| |

| |

| |

Functions of Random Variables | |

| |

| |

| |

Introduction | |

| |

| |

| |

Functions of One Random Variable | |

| |

| |

| |

Linear Functions | |

| |

| |

| |

Power Functions | |

| |

| |

| |

Expectation of a Function of One Random Variable | |

| |

| |

| |

Moments of a Linear Function | |

| |

| |

| |

Sums of Independent Random Variables | |

| |

| |

| |

Moments of the Sum of Random Variables | |

| |

| |

| |

Sum of Discrete Random Variables | |

| |

| |

| |

Sum of Independent Binomial Random Variables | |

| |

| |

| |

Sum of Independent Poisson Random Variables | |

| |

| |

| |

The Spare Parts Problem | |

| |

| |

| |

Minimum of Two Independent Random Variables | |

| |

| |

| |

Maximum of Two Independent Random Variables | |

| |

| |

| |

Comparison of the Interconnection Models | |

| |

| |

| |

Two Functions of Two Random Variables | |

| |

| |

| |

Application of the Transformation Method | |

| |

| |

| |

Laws of Large Numbers | |

| |

| |

| |

The Central Limit Theorem | |

| |

| |

| |

Order Statistics | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Problems | |

| |

| |

| |

Transform Methods | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Characteristic Function | |

| |

| |

| |

Moment-Generating Property of the Characteristic Function | |

| |

| |

| |

The s-Transform | |

| |

| |

| |

Moment-Generating Property of the s-Transform | |

| |

| |

| |

The s-Transforms of Some Well-Known PDFs | |

| |

| |

| |

The s-Transform of the PDF of the Sum of Independent Random Variables | |

| |

| |

| |

The z-Transform | |

| |

| |

| |

Moment-Generating Property of the z-Transform | |

| |

| |

| |

The z-Transform of the Bernoulli Distribution | |

| |

| |

| |

The z-Transform of the Binomial Distribution | |

| |

| |

| |

The z-Transform of the Geometric Distribution | |

| |

| |

| |

The z-Transform of the Poisson Distribution | |

| |

| |

| |

The z-Transform of the PMF of the Sum of Independent Random Variables | |

| |

| |

| |

The z-Transform of the Pascal Distribution | |

| |

| |

| |

Random Sum of Random Variables | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Problems | |

| |

| |

| |

Introduction to Random Processes | |

| |

| |

| |

Introduction | |

| |

| |

| |

Classification of Random Processes | |

| |

| |

| |

Characterizing a Random Process | |

| |

| |

| |

Mean and Autocorrelation Function of a Random Process | |

| |

| |

| |

The Autocovariance Function of a Random Process | |

| |

| |

| |

Crosscorrelation and Crosscovariance Functions | |

| |

| |

| |

Review of Some Trigonometric Identities | |

| |

| |

| |

Stationary Random Processes | |

| |

| |

| |

Strict-Sense Stationary Processes | |

| |

| |

| |

Wide-Sense Stationary Processes | |

| |

| |

| |

Ergodic Random Processes | |

| |

| |

| |

Power Spectral Density | |

| |

| |

| |

White Noise | |

| |

| |

| |

Discrete-Time Random Processes | |

| |

| |

| |

Mean, Autocorrelation Function, and Autocovariance Function | |

| |

| |

| |

Power Spectral Density | |

| |

| |

| |

Sampling of Continuous-Time Processes | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Problems | |

| |

| |

| |

Linear Systems with Random Inputs | |

| |

| |

| |

Introduction | |

| |

| |

| |

Overview of Linear Systems with Deterministic Inputs | |

| |

| |

| |

Linear Systems with Continuous-Time Random Inputs | |

| |

| |

| |

Linear Systems with Discrete-Time Random Inputs | |

| |

| |

| |

Autoregressive Moving Average Process | |

| |

| |

| |

Moving Average Process | |

| |

| |

| |

Autoregressive Process | |

| |

| |

| |

ARMA Process | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Problems | |

| |

| |

| |

Some Models of Random Processes | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Bernoulli Process | |

| |

| |

| |

Random Walk | |

| |

| |

| |

Gambler's Ruin | |

| |

| |

| |

The Gaussian Process | |

| |

| |

| |

White Gaussian Noise Process | |

| |

| |

| |

Poisson Process | |

| |

| |

| |

Counting Processes | |

| |

| |

| |

Independent Increment Processes | |

| |

| |

| |

Stationary Increments | |

| |

| |

| |

Definitions of a Poisson Process | |

| |

| |

| |

Interarrival Times for the Poisson Process | |

| |

| |

| |

Conditional and Joint PMFs for Poisson Processes | |

| |

| |

| |

Compound Poisson Process | |

| |

| |

| |

Combinations of Independent Poisson Processes | |

| |

| |

| |

Competing Independent Poisson Processes | |

| |

| |

| |

Subdivision of a Poisson Process and the Filtered Poisson Process | |

| |

| |

| |

Random Incidence | |

| |

| |

| |

Nonhomogeneous Poisson Process | |

| |

| |

| |

Markov Processes | |

| |

| |

| |

Discrete-Time Markov Chains | |

| |

| |

| |

State Transition Probability Matrix | |

| |

| |

| |

The n-Step State Transition Probability | |

| |

| |

| |

State Transition Diagrams | |

| |

| |

| |

Classification of States | |

| |

| |

| |

Limiting-State Probabilities | |

| |

| |

| |

Doubly Stochastic Matrix | |

| |

| |

| |

Continuous-Time Markov Chains | |

| |

| |

| |

Birth and Death Processes | |

| |

| |

| |

Gambler's Ruin as a Markov Chain | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Problems | |

| |

| |

| |

Introduction to Statistics | |

| |

| |

| |

Introduction | |

| |

| |

| |

Sampling Theory | |

| |

| |

| |

The Sample Mean | |

| |

| |

| |

The Sample Variance | |

| |

| |

| |

Sampling Distributions | |

| |

| |

| |

Estimation Theory | |

| |

| |

| |

Point Estimate, Interval Estimate, and Confidence Interval | |

| |

| |

| |

Maximum Likelihood Estimation | |

| |

| |

| |

Minimum Mean Squared Error Estimation | |

| |

| |

| |

Hypothesis Testing | |

| |

| |

| |

Hypothesis Test Procedure | |

| |

| |

| |

Type I and Type II Errors | |

| |

| |

| |

One-Tailed and Two-Tailed Tests | |

| |

| |

| |

Curve Fitting and Linear Regression | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Problems | |

| |

| |

| |

Table for the CDF of the Standard Normal Random Variable | |

| |

| |

Bibliography | |

| |

| |

Index | |