Probability and Statistics

This book is designed for engineering students studying for the core paper on probability and statistics The topics have been dealt in a coherent manner supported by illustrations for better compre nothension. Each chapter is replete with examples and exercises. The book also has numerous Multiple Choice Questions at the end of each chapter thus providing the student with an abundant repository of exam specific problems. nbsp Probability and Stastics 1 Roadmap to the Syllabus 7 Contents 8 List of Symbols 16 Chapter 1 Probability 20 1.1 Introduction 20 1.2 Sets and Set Operations 21 1.3 Principle of Counting 25 1.4 Permutations and Combinations 26 1.5 Binomial Expansion 29 1.6 Introduction to Probability 29 1.7 Axioms of ­Probability 31 1.8 Basic Theorems 31 1.9 Conditional Probability and Independent Events 34 1.10 Theorem of Total Probability (or the Rule of Elimination) 39 1.11 Bayes’ Theorem or Rule 41 Exercises 45 Multiple Choice Questions 48 Fill in the Blanks 51 Chapter 2 Probability Distribution 54 2.1 Introduction 54 2.2 Random Variables 54 2.3 Probability Distribution 55 2.4 Expectation or Mean or Expected Value 63 2.5 Variance and Standard Deviation 63 2.6 Probability Density Functions 66 2.7 Chebyshev’s Theorem 70 Exercises 74 Fill in the Blanks 75 Chapter 3 Special Distribution 77 3.1 Introduction 77 3.2 Binomial (Bernoulli) Distribution 77 3.3 Poisson Distribution 80 3.4 Uniform Distribution 85 3.5 Exponential Distribution 86 3.6 Normal Distribution 88 Exercises 99 Multiple Choice Questions 104 Fill in the Blanks 109 Chapter 4 Sampling Distributions 112 4.1 Introduction 112 4.2 Population and Sample 112 4.3 Sampling Distribution 113 4.4 Sampling Distribution of Means (σ Known) 114 4.5 Sampling Distribution of ­Proportions 119 4.6 Sampling Distribution of Differences and Sums 121 4.7 Sampling Distribution of Means (σ Unknown): t-Distribution 126 4.8 Chi-square (χ2) Distribution 129 4.9 Sampling Distribution of Variance s 2 131 4.10 Snedecor’s F-Distribution 133 4.11 Fisher’s z-Distribution 134 Exercises 135 Multiple Choice Questions 136 Fill in the Blanks 138 Chapter 5 Estimation Theory 139 5.1 Introduction 139 5.2 Statistical Inference 139 5.3 Point Estimation 139 5.4 Interval Estimation 146 5.5 Bayesian Estimation 149 Exercises 151 Fill in the Blanks 152 Chapter 6 Inferences Concerning Means and Proportions 154 6.1 Introduction 154 6.2 Statistical Hypotheses 154 6.3 Tests of Hypotheses and ­Significance 154 6.4 Type I and Type II Errors 155 6.5 Levels of Significance 155 6.6 Statistical Test of Hypothesis Procedure 156 6.7 Reasoning of Statistical Test of Hypothesis 158 6.8 Inference Concerning Two Means 166 Exercises 169 Fill in the Blanks 170 Chapter 7 Tests of Significance 172 7.1 Introduction 172 7.2 Test for One Mean (Small Sample) 172 7.3 Test for Two Means 178 7.4 Test of Hypothesis 184 7.5 Analysis of r × c Tables (Contingency Tables) 189 7.6 Goodness-of-Fit Test: χ2 ­Distribution 190 7.7 Estimation of Proportions 193 Exercises 196 Fill in the Blanks 199 Chapter 8 Curve Fitting: Regression and Correlation Analysis 202 8.1 Introduction 202 8.2 Linear Regression 202 8.3 Regression Analysis 206 8.4 Inferences Based on Least Squares Estimation 206 8.5 Multiple Regression 212 8.6 Correlation Analysis 215 8.7 Least Squares Line in Terms of Sample Variances and Covariance 217 8.8 Standard Error of Estimate 218 8.9 Spearman’s Rank Correlation 225 8.10 Correlation for Bivariate Frequency Distribution 228 Exercises 232 Fill in the Blanks 235 Chapter 9 Analysis of Variance 240 9.1 Analysis of Variance (ANOVA) 240 9.2 What is ANOVA? 240 9.3 The Basic Principle 0f Anova 241 9.4 Anova Technique 241 9.5 Setting Up Analysis of Variance Table 243 9.6 Shortcut Method For One-Way Anova 243 9.7 Coding Method 244 9.8 Two-Way Anova 247 9.9 Anova in Latin-Square Design 253 Exercises 256 Chapter 10 Statistical Quality Control 258 10.1 Properties of Control Charts 259 10.2 Shewhart Control Charts for Measurements 263 10.3 Shewhart Control Charts for Attributes 269 10.4 Tolerance Limits 274 10.5 Acceptance Sampling 276 10.6 Two-stage Acceptance Sampling 280 Exercises 283 Chapter 11 Queueing Theory 290 11.1 Introduction 290 11.2 Queues or Waiting Lines 290 11.3 Elements of a Basic Queueing System 290 11.4 Description of a Queueing System 292 11.5 Classification of Queueing Systems 292 11.6 Queueing Problem 294 11.7 States of Queueing Theory 294 11.8 Probability Distribution in Queueing Systems 295 11.9 Kendall’s Notation for ­Representing Queueing ­Models 304 11.10 Basic Probabilistic Queueing Models 304 Exercises 320 Fill in the Blanks 323 Appendix A: Test Based on Normal Distributions 326 Appendix B: Statistical Tables 338 Appendix C: Basic Results 362 Additional Solved Problems 374 Index 394 This book is designed for engineering students studying the core paper on probability and statistics during their second or third years. It includes detailed explanation of theory with numerous examples and exercises, as well as relevant references to engineering applications. Each also has numerous objective type questions and answers and hints are provided for all the exercise problems and objective type questions