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Probability, Statistics and Random Processes

Book Description

Probability, Statistics and Random Processes is designed to meet the requirements of students and is intended for beginners to help them understand the concepts from the first principles. Spread across 16 chapters, it discusses the theoretical aspects that have been refined and updated to reflect the current developments in the subjects. It expounds on theoretical concepts that have immense practical applications, giving adequate proofs to establish significant theorems.

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. Preface
  5. Chapter 1: Probability
    1. Introduction
    2. 1.1 Elementary Concepts of Set Theory
    3. 1.2 Permutations and Combinations
    4. 1.3 Introduction of Probability
    5. 1.4 Axioms of Probability
    6. 1.5 Some Elementary Results
    7. 1.6 Conditional Probability
    8. 1.7 Theorem of Total Probability
    9. 1.8 Baye’s Theorem
    10. Definitions at a Glance
    11. Formulae at a Glance
    12. Objective Type Questions
  6. Chapter 2: Random Variables (Discrete and Continuous)
    1. Introduction
    2. 2.1 Random Variable
    3. 2.2 Probability Mass Function (PMF)
    4. 2.3 Probability Density Function (PDF)
    5. 2.4 Joint Probability Distributions
    6. 2.5 Joint Density Function F(X, Y)
    7. 2.6 Stochastic Independence
    8. 2.7 Transformation of One-Dimensional Random Variable
    9. 2.8 Transformation of Two-Dimensional Random Variable
    10. Definitions at a Glance
    11. Formulae at a Glance
    12. Objective Type Questions
  7. Chapter 3: Mathematical Expectation
    1. Introduction
    2. 3.1 Mathematical Expectation
    3. 3.2 Variance
    4. 3.3 Expectation of a Function of Random Variables
    5. 3.4 Variance for Joint Distributions
    6. 3.5 Covariance
    7. 3.6 Conditional Expectation
    8. 3.7 Chebychev’s Inequality
    9. 3.8 Moments
    10. 3.9 Moment Generating Function
    11. 3.10 Characteristic Function
    12. Definitions at a Glance
    13. Formulae at a Glance
    14. Objective Type Questions
  8. Chapter 4: Standard Discrete Distributions
    1. Introduction
    2. 4.1 Binomial Distribution
    3. 4.2 Poisson Distribution
    4. 4.3 Negative Binomial Distribution
    5. 4.4 Geometric Distribution
    6. 4.5 Hyper Geometric Distribution
    7. 4.6 Uniform Distribution
    8. Definitions at a Glance
    9. Formulae at a Glance
    10. Objective Type Questions
  9. Chapter 5: Standard Continuous Distributions
    1. Introduction
    2. 5.1 Normal Distribution
    3. 5.2 Exponential Distribution
    4. 5.3 Gamma Distribution
    5. 5.4 Weibull Distribution
    6. 5.5 Central Limit Theorem
    7. Definitions at a Glance
    8. Formulae at a Glance
    9. Objective Type Questions
  10. Chapter 6: Sampling Theory and Distribution
    1. Introduction
    2. 6.1 Some Definitions
    3. 6.2 Types of Sampling
    4. 6.3 Advantages of Sampling
    5. 6.4 Sampling Distribution of a Statistic
    6. 6.5 Standard Error
    7. 6.6 Importance of Standard Error
    8. 6.7 Sampling from Normal and Non-Normal Populations
    9. 6.8 Finite Population Correction (FPC) Factor
    10. 6.9 Sampling Distribution of Means
    11. 6.10 When Population Variance is Unknown
    12. 6.11 Sampling Distribution of the Difference between Two Means
    13. 6.12 Sampling Distribution of Variance
    14. 6.13 The Chi-Square Distribution
    15. 6.14 The Student’s t-Distribution
    16. 6.15 F-Distribution
    17. Definitions at a Glance
    18. Formulae at a Glance
    19. Objective Type Questions
  11. Chapter 7: Testing of Hypothesis (Large Samples)
    1. Introduction
    2. 7.1 Statistical Hypothesis
    3. 7.2 Tests of Significance
    4. 7.3 Some Important Definitions
    5. 7.4 Steps Involved in Testing of Hypothesis
    6. 7.5 Tests of Significance
    7. Definitions at a Glance
    8. Formulae at a Glance
    9. Objective Type Questions
  12. Chapter 8: Test of Hypothesis (Small Samples)
    1. Introduction
    2. 8.1 Student’s t-Distribution
    3. 8.2 Critical Values of t
    4. 8.3 t-Test for Single Mean
    5. 8.4 t-Test for Difference of Means
    6. 8.5 Paired t-Test for Difference of Means
    7. 8.6 Snedecor’s F-Distribution
    8. 8.7 Chi-Square Distribution
    9. 8.8 Test for Independence of Attributes
    10. Definitions at a Glance
    11. Formulae at a Glance
    12. Objective Type Questions
  13. Chapter 9: Estimation
    1. Introduction
    2. 9.1 Point Estimation
    3. 9.2 Characteristics of Estimators
    4. 9.3 Interval Estimation
    5. 9.4 Confidence Interval
    6. 9.5 Some Results
    7. 9.6 Confidence Interval for Difference between Two Means (Known Variances)
    8. 9.7 Confidence Interval for Difference between Two Means (Unknown Variances)
    9. 9.8 Confidence Interval for Difference of Means (Unknown and Unequal Variances)
    10. 9.9 Confidence Interval for Difference between Means for Paired observations
    11. 9.10 Confidence Interval for Estimating the Variance
    12. 9.11 Confidence Interval for Estimating the Ratio of Two Variances
    13. 9.12 Bayesian Estimation
    14. Definitions at a Glance
    15. Formulae at a Glance
    16. Objective Type Questions
  14. Chapter 10: Curve Fitting
    1. Introduction
    2. 10.1 The Method of Least Squares
    3. 10.2 Fitting of a Straight Line
    4. 10.3 Fitting of a Second Degree Parabola
    5. 10.4 Fitting of Exponential Curve and Power Curve
    6. Definitions at a Glance
    7. Formulae at a Glance
    8. Objective Type Questions
  15. Chapter 11: Correlation
    1. Introduction
    2. 11.1 Types of Correlation
    3. 11.2 Methods of Correlation
    4. 11.3 Properties of Correlation Coefficient
    5. 11.4 Coefficient of Correlation for Grouped Data
    6. 11.5 Rank Correlation
    7. 11.6 Limitations of Spearman’s Correlation Coefficient Method
    8. 11.7 Tied Ranks
    9. 11.8 Concurrent Deviations Method
    10. Definitions at a Glance
    11. Formulae at a Glance
    12. Objective Type Questions
  16. Chapter 12: Regression
    1. 12.1 Regression
    2. 12.2 Lines of Regression
    3. 12.3 Regression Coefficients
    4. 12.4 Difference between Regression and Correlation Analysis
    5. 12.5 Angle between Two Lines of Regression
    6. 12.6 Standard Error of Estimate
    7. 12.7 Limitations of Regression Analysis
    8. 12.8 Regression Curves
    9. Definitions at a Glance
    10. Formulae at a Glance
    11. Objective Type Questions
  17. Chapter 13: Queuing Theory
    1. Introduction
    2. 13.1 Elements of a Queuing Model
    3. 13.2 Distribution of Inter-Arrival Time
    4. 13.3 Distribution of Service Time
    5. 13.4 Queuing Process
    6. 13.5 Transient State and Steady State
    7. 13.6 Some Notations
    8. 13.7 Probability Distributions in Queuing System
    9. 13.8 Pure Birth Process
    10. 13.9 Pure Death Process
    11. 13.10 Classification of Queuing Models: (Single Server Queuing Models)
    12. 13.11 Multi-Server Queuing Models
    13. Definitions at a Glance
    14. Formulae at a Glance
    15. Objective Type Questions
  18. Chapter 14: Design of Experiments
    1. Introduction
    2. 14.1 Assumptions of Analysis of Variance
    3. 14.2 One-Way Classification
    4. 14.3 The Analysis from Decomposition of the Individual Observations
    5. 14.4 Two-Way Classification
    6. 14.5 Completely Randomized Design (CRD)
    7. 14.6 Latin Square Design (LSD)
    8. 14.7 Randomized Block Design (RBD)
    9. Definitions at a Glance
    10. Formulae at a Glance
    11. Objective Type Questions
  19. Chapter 15: Random Process
    1. Introduction
    2. 15.1 Classification of Random Processes
    3. 15.2 Stationarity
    4. 15.3 Second Order Stationary Process
    5. 15.4 Wide Sense Stationary Process
    6. 15.5 Cross Correlation Function
    7. 15.6 Statistical Averages
    8. 15.7 Time Averages
    9. 15.8 Statistical Independence
    10. 15.9 Ergodic Random Process
    11. 15.10 Mean-Ergodic Theorem
    12. 15.11 Correlation Ergodic Process
    13. 15.12 Correlation Functions
    14. 15.13 Covariance Functions
    15. 15.14 Spectral Representation
    16. 15.15 Discrete Time Processes
    17. 15.16 Discrete Time Sequences
    18. 15.17 Some Noise Definitions
    19. 15.18 Types of Noise
    20. Definitions at a Glance
    21. Formulae at a Glance
    22. Objective Type Questions
  20. Chapter 16: Advanced Random Process
    1. Introduction
    2. 16.1 Poisson Process
    3. 16.2 Mean and Auto Correlation of the Poisson Process
    4. 16.3 Markov Process
    5. 16.4 Chapman-Kolmogorov Theorem
    6. 16.5 Definitions in Markov Chain
    7. 16.6 Application to the Theory of Queues
    8. 16.7 Random Walk
    9. 16.8 Gaussian Process
    10. 16.9 Band Pass Process
    11. 16.10 Narrow Band Gaussian Process
    12. 16.11 Band Limited Process
    13. Definitions at a Glance
    14. Formulae at a Glance
    15. Objective Type Questions
  21. Appendix A
  22. Appendix B
  23. Appendix C
  24. Appendix D
  25. Notes
  26. Acknowledgements
  27. Copyright
  28. Back Cover