Foundations of Probability and Statistics

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1 Foundations of Probability and Statistics William C. Rinaman Le Moyne College Syracuse, New York Saunders College Publishing Harcourt Brace College Publishers Fort Worth Philadelphia San Diego New York Orlando Austin San Antonio Toronto Montreal London Sydney Tokyo

Table of Contents Chapter 1 Random Experiments and Probability 1 1.1 Introduction 1 1.2 Set Theory 3 1.3 Random Experiments 12 1.4 The Axioms of Probability 17 1.5 Counting Elementary Events 27 1.

2 Table of Contents Chapter 1 Random Experiments and Probability Introduction Set Theory Random Experiments The Axioms of Probability Counting Elementary Events Probabilities for Finite Sample Spaces with Equally Likely Outcomes 41 Chapter Summary 53 Review Exercises 56 Chapter 2 Conditional Probability and Independence Introduction Conditional Probability Bayes'Rule Independence 77 Chapter Summary 92 Review Exercises 92 Chapter 3 Random Variables Introduction 95 vü 3.2 The Distribution Function 99

viii Table of Contents 3.3 Discrete Random Variables 108 3.4 Continuous Random Variables 114 3.5 Random Vectors 121 3.6 Independent Random Variables 139 3.

3 viii Table of Contents 3.3 Discrete Random Variables Continuous Random Variables Random Vectors Independent Random Variables Conditional Distributions 145 Chapter Summary 149 Review Exercises 152 Chapter 4 Expectation Introduction Expected Value Expectation of a Function of a Random Variable Moments Moment Generating Functions Conditional Expectation 197 Chapter Summary 203 Review Exercises 204 Chapter 5 Probability Distributions Introduction Hypergeometric Distribution Distributions Based on Independent Bernoulli Trials Multinomial Distribution Poisson Distribution Uniform Distribution Normal Distribution Gamma Distribution Other Continuous Distributions Weibull Distribution Beta Distribution Cauchy Distribution Double Exponential Distribution Logistic Distribution Bivariate Normal Distribution 250

Table of Contents ix Chapter Summary 253 Review Exercises 254 Chapter 6 Distributions of Functions of Random Variables 257 6.1 Introduction 257 6.2 Transformations of Discrete Random Variables 257 6.

4 Table of Contents ix Chapter Summary 253 Review Exercises 254 Chapter 6 Distributions of Functions of Random Variables Introduction Transformations of Discrete Random Variables The Distribution Function Method The Transformation of Variables Method Moment Generating Function Method Sums of Independent Random Variables Distribution of Functions of Several Random Variables 277 Chapter Summary 286 Review Exercises 288 Chapter 7 Limit Theorems Introduction Convergence in Probability Central Limit Theorem More on Convergence in Distribution 315 Chapter Summary 319 Review Exercises 322 Chapter 8 Statistical Models Introduction Populations and Random Sampling Numerical Summaries Sample Mean Sample Median Sample Variance Median Absolute Deviation Percentile-Based Measures of Dispersion Sample Skewness Sample Kurtosis An Example 338

x Table of Contents 8.4 Graphical Summaries 341 8.4.1 Quantile-Quantile Plots 342 8.4.2 Histograms 345 8.4.3 Stem-and-Leaf Plots 346 Chapter Summary 349 Review Exercises 350 Chapter 9 Sampling Distributions 9.

5 x Table of Contents 8.4 Graphical Summaries Quantile-Quantile Plots Histograms Stem-and-Leaf Plots 346 Chapter Summary 349 Review Exercises 350 Chapter 9 Sampling Distributions 9.1 Introduction Distributions Based on Samples from Normal Populations 9.3 Independence of X and s Order Statistics 362 Chapter Summary 377 Review Exercises 378 Chapter 10 Point Estimation 10.1 Introduction Maximum Likelihood Estimation Method of Moments Properties of Estimators Unbiasedness Efficiency Sufflciency Minimal Sufficient Statistics Completeness Large Sample Properties of Estimators Consistency Asymptotic Efficiency Bayes Estimators Robust Estimation 441 Chapter Summary 444 Review Exercises 449

Table of Contents xi Chapter 11 Confidence Intervals 451 11.1 Introduction 451 11.2 Pivotal Quantity Method 453 11.3 Method Based on Sampling Distributions 463 11.

6 Table of Contents xi Chapter 11 Confidence Intervals Introduction Pivotal Quantity Method Method Based on Sampling Distributions Large Sample Confidence Intervals A Nonparametnc Confidence Interval 472 Chapter Summary 474 Review Exercises 476 Chapter 12 Hypothesis Testing Introduction Types of Errors Testing Simple Hypotheses Uniformly Most Powerful Tests Likelihood Ratio Tests Chi-Square Tests Large Sample Tests Sequential Probability Ratio Test Nonparametric Tests Sign Test Wilcoxon Signed Rank Test Wilcoxon Rank Sum Test 553 Chapter Summary 558 Review Exercises 564 Chapter 13 Regression and Correlation Introduction Simple Linear Regression Matrix Calculus Multiple Regression Correlation Nonparametric Methods 615 Chapter Summary 622 Review Exercises 630

- xii Table of Contents Chapter 14 Appendix A Analysis of Variance 14.1 Introduction 633 14.2 Experimental Design 634 14.3 One-Way Analysis of Variance 14.4 Two-Way Analysis of Variance 14.

7 - xii Table of Contents Chapter 14 Appendix A Analysis of Variance 14.1 Introduction Experimental Design One-Way Analysis of Variance 14.4 Two-Way Analysis of Variance 14.5 Estimation Nonparametric Methods 673 Chapter Summary 681 Review Exercises 687 Statistical Tables A.1 I Binomial Distribution A.l II Poisson Distribution A.7 III Normal Distribution A.l3 IV Chi-Square Distribution A.14 V Student's t Distribution A.l 5 VI F Distribution A.l6 VII Wilcoxon Signed Rank Test A.20 VIII Wilcoxon Rank Sum Test A.27 IX Spearman's Rho A.35 X Kruskal-Wallis Test A.38 XI Friedman Test A.53 Appendix B Probability Distributions Discrete Random Variables A.59 Continuous Random Variables A.61 A.59 Appendix C Bibliography A.63 Appendix D Answers to Selected Odd-Numbered Exercises A.65 Index J LI

Random Experiments and Probability Most people know intuitively what the word "probability" means. Many of us have encountered probabilities in our daily activities.

8 Random Experiments and Probability Most people know intuitively what the word "probability" means. Many of us have encountered probabilities in our daily activities. For example, if we flip a coin it is our understanding that heads should result about half the time. When a local weather forecaster states that there is a 40 percent chance for rain tomorrow we have a general understanding that what is meant is that, given the current conditions, rain may occur 4 times out of 10. If we play bridge, we may draw on past experience to assess the likelihood that the cards held by our opponents have been distributed in a certain way in order to plan how we play a current hand. In fact, anyone who is consistently successful at games involving chance such as poker and backgammon is well aware of the relative frequency of the various possible outcomes in the play of the game. The situations mentioned above have some things in common. In each we face a Situation that is, at least in principle, well defined and repeatable. The possible outcomes are known, but the particular outcome that will take place this time cannot be predicted. What probability attempts to do is to determine a numerical value that teils what proportion of the time each possibility will occur. One could correctly argue that the weather example is not like the others. It does not involve chance occurrences but rather represents a less than complete understanding of the physics of the atmosphere. This is the case in a number of areas where probability may be applied. Many physical situations are so complicated that it is not feasible to develop a mathematical model that accounts for all of the variables. In such cases it is common to develop modeis that use the main variables and then lump together the unused variables in a term that represents the unpredicted part of the model called the "noise." The noise in the model is what injects randomness into predictions.

Institute of Actuaries of India

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2018 Examinations Subject CT3 Probability and Mathematical Statistics Core Technical Syllabus 1 June 2017 Aim The