William Turin

Probability, random processes, and statistical analysis

1. Introduction Part I. Probability, Random Variables and Statistics: 2. Probability 3. Discrete random variables 4. Continuous random variables 5. Functions of random variables and their distributions 6. Fundamentals of statistical analysis 7. Distributions derived from the normal distribution Part II. Transform Methods, Bounds and Limits: 8. Moment generating function and characteristic function 9. Generating function and Laplace transform 10. Inequalities, bounds and large deviation approximation 11. Convergence of a sequence of random variables, and the limit theorems Part III. Random Processes: 12. Random process 13. Spectral representation of random processes and time series 14. Poisson process, birth-death process, and renewal process 15. Discrete-time Markov chains 16. Semi-Markov processes and continuous-time Markov chains 17. Random walk, Brownian motion, diffusion and ito processes Part IV. Statistical Inference: 18. Estimation and decision theory 19. Estimation algorithms Part V. Applications and Advanced Topics: 20. Hidden Markov models and applications 21. Probabilistic models in machine learning 22. Filtering and prediction of random processes 23. Queuing and loss models.

Probability, Random Processes, and Statistical Analysis: Distributions derived from the normal distribution

In Sections 4.2.4 and 4.3.1 we defined the normal (or Gaussian) distributions for both single and multiple variables and discussed their properties. The normal distribution plays a central role in the mathematical theory of statistics for at least two reasons. First, the normal distribution often describes a variety of physical quantities observed in the real world. In a communication system, for example, a received waveform is often a superposition of a desired signal waveform and (unwanted) noise process, and the amplitude of the noise is often normally distributed, because the source of such noise is usually what is known as thermal noise at the receiver front. The normality of thermal noise is a good example of manifestation in the real world of the CLT, which says that the sum of a large number of independent RVs, properly scaled, tends to be normally distributed. In Chapter 3 we saw that the binomial distribution and the Poisson distribution also tend to a normal distribution in the limit. We also discussed the CLT and asymptotic normality. The second reason for the frequent use of the normal distribution is its mathematical tractability. For instance, sums of independent normal RVs are themselves normally distributed. Such reproductivity of the distribution is enjoyed only by a limited class of distributions (that is, binomial, gamma, Poisson). Many important results in the theory of statistics are founded on the assumption of a normal distribution.

Probability, Random Processes, and Statistical Analysis: Probability, random variables, and statistics

1. Introduction Part I. Probability, Random Variables and Statistics: 2. Probability 3. Discrete random variables 4. Continuous random variables 5. Functions of random variables and their distributions 6. Fundamentals of statistical analysis 7. Distributions derived from the normal distribution Part II. Transform Methods, Bounds and Limits: 8. Moment generating function and characteristic function 9. Generating function and Laplace transform 10. Inequalities, bounds and large deviation approximation 11. Convergence of a sequence of random variables, and the limit theorems Part III. Random Processes: 12. Random process 13. Spectral representation of random processes and time series 14. Poisson process, birth-death process, and renewal process 15. Discrete-time Markov chains 16. Semi-Markov processes and continuous-time Markov chains 17. Random walk, Brownian motion, diffusion and ito processes Part IV. Statistical Inference: 18. Estimation and decision theory 19. Estimation algorithms Part V. Applications and Advanced Topics: 20. Hidden Markov models and applications 21. Probabilistic models in machine learning 22. Filtering and prediction of random processes 23. Queuing and loss models.

Probability, Random Processes, and Statistical Analysis: List of abbreviations and acronyms

1. Introduction Part I. Probability, Random Variables and Statistics: 2. Probability 3. Discrete random variables 4. Continuous random variables 5. Functions of random variables and their distributions 6. Fundamentals of statistical analysis 7. Distributions derived from the normal distribution Part II. Transform Methods, Bounds and Limits: 8. Moment generating function and characteristic function 9. Generating function and Laplace transform 10. Inequalities, bounds and large deviation approximation 11. Convergence of a sequence of random variables, and the limit theorems Part III. Random Processes: 12. Random process 13. Spectral representation of random processes and time series 14. Poisson process, birth-death process, and renewal process 15. Discrete-time Markov chains 16. Semi-Markov processes and continuous-time Markov chains 17. Random walk, Brownian motion, diffusion and ito processes Part IV. Statistical Inference: 18. Estimation and decision theory 19. Estimation algorithms Part V. Applications and Advanced Topics: 20. Hidden Markov models and applications 21. Probabilistic models in machine learning 22. Filtering and prediction of random processes 23. Queuing and loss models.

Probability, Random Processes, and Statistical Analysis: Frontmatter

1. Introduction Part I. Probability, Random Variables and Statistics: 2. Probability 3. Discrete random variables 4. Continuous random variables 5. Functions of random variables and their distributions 6. Fundamentals of statistical analysis 7. Distributions derived from the normal distribution Part II. Transform Methods, Bounds and Limits: 8. Moment generating function and characteristic function 9. Generating function and Laplace transform 10. Inequalities, bounds and large deviation approximation 11. Convergence of a sequence of random variables, and the limit theorems Part III. Random Processes: 12. Random process 13. Spectral representation of random processes and time series 14. Poisson process, birth-death process, and renewal process 15. Discrete-time Markov chains 16. Semi-Markov processes and continuous-time Markov chains 17. Random walk, Brownian motion, diffusion and ito processes Part IV. Statistical Inference: 18. Estimation and decision theory 19. Estimation algorithms Part V. Applications and Advanced Topics: 20. Hidden Markov models and applications 21. Probabilistic models in machine learning 22. Filtering and prediction of random processes 23. Queuing and loss models.

Probability, Random Processes, and Statistical Analysis: Index

1. Introduction Part I. Probability, Random Variables and Statistics: 2. Probability 3. Discrete random variables 4. Continuous random variables 5. Functions of random variables and their distributions 6. Fundamentals of statistical analysis 7. Distributions derived from the normal distribution Part II. Transform Methods, Bounds and Limits: 8. Moment generating function and characteristic function 9. Generating function and Laplace transform 10. Inequalities, bounds and large deviation approximation 11. Convergence of a sequence of random variables, and the limit theorems Part III. Random Processes: 12. Random process 13. Spectral representation of random processes and time series 14. Poisson process, birth-death process, and renewal process 15. Discrete-time Markov chains 16. Semi-Markov processes and continuous-time Markov chains 17. Random walk, Brownian motion, diffusion and ito processes Part IV. Statistical Inference: 18. Estimation and decision theory 19. Estimation algorithms Part V. Applications and Advanced Topics: 20. Hidden Markov models and applications 21. Probabilistic models in machine learning 22. Filtering and prediction of random processes 23. Queuing and loss models.

Probability, Random Processes, and Statistical Analysis: Statistical inference

1. Introduction Part I. Probability, Random Variables and Statistics: 2. Probability 3. Discrete random variables 4. Continuous random variables 5. Functions of random variables and their distributions 6. Fundamentals of statistical analysis 7. Distributions derived from the normal distribution Part II. Transform Methods, Bounds and Limits: 8. Moment generating function and characteristic function 9. Generating function and Laplace transform 10. Inequalities, bounds and large deviation approximation 11. Convergence of a sequence of random variables, and the limit theorems Part III. Random Processes: 12. Random process 13. Spectral representation of random processes and time series 14. Poisson process, birth-death process, and renewal process 15. Discrete-time Markov chains 16. Semi-Markov processes and continuous-time Markov chains 17. Random walk, Brownian motion, diffusion and ito processes Part IV. Statistical Inference: 18. Estimation and decision theory 19. Estimation algorithms Part V. Applications and Advanced Topics: 20. Hidden Markov models and applications 21. Probabilistic models in machine learning 22. Filtering and prediction of random processes 23. Queuing and loss models.

Probability, Random Processes, and Statistical Analysis: Contents

1. Introduction Part I. Probability, Random Variables and Statistics: 2. Probability 3. Discrete random variables 4. Continuous random variables 5. Functions of random variables and their distributions 6. Fundamentals of statistical analysis 7. Distributions derived from the normal distribution Part II. Transform Methods, Bounds and Limits: 8. Moment generating function and characteristic function 9. Generating function and Laplace transform 10. Inequalities, bounds and large deviation approximation 11. Convergence of a sequence of random variables, and the limit theorems Part III. Random Processes: 12. Random process 13. Spectral representation of random processes and time series 14. Poisson process, birth-death process, and renewal process 15. Discrete-time Markov chains 16. Semi-Markov processes and continuous-time Markov chains 17. Random walk, Brownian motion, diffusion and ito processes Part IV. Statistical Inference: 18. Estimation and decision theory 19. Estimation algorithms Part V. Applications and Advanced Topics: 20. Hidden Markov models and applications 21. Probabilistic models in machine learning 22. Filtering and prediction of random processes 23. Queuing and loss models.

Probability, Random Processes, and Statistical Analysis: Applications and advanced topics

1. Introduction Part I. Probability, Random Variables and Statistics: 2. Probability 3. Discrete random variables 4. Continuous random variables 5. Functions of random variables and their distributions 6. Fundamentals of statistical analysis 7. Distributions derived from the normal distribution Part II. Transform Methods, Bounds and Limits: 8. Moment generating function and characteristic function 9. Generating function and Laplace transform 10. Inequalities, bounds and large deviation approximation 11. Convergence of a sequence of random variables, and the limit theorems Part III. Random Processes: 12. Random process 13. Spectral representation of random processes and time series 14. Poisson process, birth-death process, and renewal process 15. Discrete-time Markov chains 16. Semi-Markov processes and continuous-time Markov chains 17. Random walk, Brownian motion, diffusion and ito processes Part IV. Statistical Inference: 18. Estimation and decision theory 19. Estimation algorithms Part V. Applications and Advanced Topics: 20. Hidden Markov models and applications 21. Probabilistic models in machine learning 22. Filtering and prediction of random processes 23. Queuing and loss models.

Probability, Random Processes, and Statistical Analysis: Acknowledgments

1. Introduction Part I. Probability, Random Variables and Statistics: 2. Probability 3. Discrete random variables 4. Continuous random variables 5. Functions of random variables and their distributions 6. Fundamentals of statistical analysis 7. Distributions derived from the normal distribution Part II. Transform Methods, Bounds and Limits: 8. Moment generating function and characteristic function 9. Generating function and Laplace transform 10. Inequalities, bounds and large deviation approximation 11. Convergence of a sequence of random variables, and the limit theorems Part III. Random Processes: 12. Random process 13. Spectral representation of random processes and time series 14. Poisson process, birth-death process, and renewal process 15. Discrete-time Markov chains 16. Semi-Markov processes and continuous-time Markov chains 17. Random walk, Brownian motion, diffusion and ito processes Part IV. Statistical Inference: 18. Estimation and decision theory 19. Estimation algorithms Part V. Applications and Advanced Topics: 20. Hidden Markov models and applications 21. Probabilistic models in machine learning 22. Filtering and prediction of random processes 23. Queuing and loss models.