Introduction to Probability and Statistics

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This book includes the probability of events, discrete random variables and their distribution, continuous random variables and their distribution, digital characteristics of random variables, law of large numbers and central limit theorem, sampling distribution, parameter estimation and hypothesis testing.All of the authors of this book have the background of visiting British and American university. The writing language is easy to understand, and the content has moderate difficulty.This book can be used for the teaching of probability and statistics courses of Sino-foreign cooperation projects and foreign student programs in universities of science and engineering (non-mathematics majors), as well as bilingual teaching of probability and statistics.
Table of Contents
Chapter 1Introduction001
1.1The Origin of Probability Theory and Mathematical Statistics001
1.2Random Phenomena and Random Trials002
1.3Statistical Regularity of Random Phenomena003
1.4Some Important Applications of Probability and Statistics004
Chapter 2Basic Probability006
2.1Set Theory006
2.1.1Sets, Elements, and Subsets006
2.1.2Set Operation: Union, Intersection, Complement and Set Differences, Exclusive and Opposite008
2.1.3Experiments, Sample Spaces, and Events010
2.2Set Functions011
2.2.1Boolean Algebras011
2.2.3Examples of Measures013
2.2.4Measures on Partitions of Sets014
2.3Probability as Measure014
2.3.1Properties of Probability015
2.4Assigning Probabilities016
2.4.1Classical Probability Based on Symmetry016
2.4.2Counting Methods for Classical Probability: Permutations and Combinations017
2.4.3Estimated Probability(Relative Frequency)019
2.4.4Subjective Probabilities020
2.5Conditional Probability021
2.5.2The Law of Total Probability023
2.5.3Bayes' Theorem024
Chapter 3Discrete Random Variables028
3.1Random Variables028
3.2Probability Distributions for Discrete Random Variables030
3.2.1Probability Mass Function (PMF)030
3.2.2Cumulative Distribution Function(CDF)031
3.2.3Derived Distributions of Discrete Random Variables034
3.3Some Important Discrete Probability Distributions035
3.3.1The Bernoulli Distribution035
3.3.2The Binomial Distribution036
3.3.3Hypergeometric Distributions037
3.3.4The Poisson Distribution040
3.4Multiple Discrete Random Variables042
3.4.1Joint Distribution 042
3.4.2Marginal Distribution 044
3.4.3Conditional Distribution045
3.4.4Independence of Discrete Random Variables048
3.4.5Derived Distributions of Multiple Discrete Random Variables049
Chapter 4Continuous Random Variables054
4.1Continuous Random Variable054
4.1.1Continuous Probability Distribution 054
4.1.2Some Important Continuous Distribution059
4.2Multiple Continuous Random Variables066
4.2.1Joint Distribution066
4.2.2Marginal Distribution067
4.2.3Conditional Distribution069
4.2.4Independence of Continuous Random Variables070
4.3Derived Distributions of Continuous Variable071
Chapter 5Numerical Characteristics of Random Variables080
5.1.1Average & Expectation080
5.1.2Expectations for Functions of Random Variables 082
5.1.3Moments of the Random Variable085
5.2Variance 086
5.2.1Variance & Standard Deviation086
5.2.2Expectations & Variance for Several Common Distributions091
5.3Covariance and Correlation Coefficient094
5.3.1Covariance and Correlation Coefficient094
5.3.2The Essence of Covariance and Correlation Coefficient097
Chapter 6Sums of Random Variables105
6.1Sums of Independent and Identically Distributed Random Variables105
6.2Laws of Large Numbers107
6.2.1Chebyshev's Inequality107
6.2.2The Weak Law of Large Numbers107
6.3The Central Limit Theorem (CLT)108
6.3.1Example: Sums of Exponential Random Variables109
6.3.2Example: Sums of Bernoulli Random Variables, and the Normal Approximation to the Binomial Distribution109
Chapter 7Random Samples and Sampling Distributions113
7.1Random Sampling113
7.2Some Important Statistics115
7.2.1Location Measures of a Sample116
7.2.2Variability Measures of a Sample117
7.3Sampling Distributions119
7.4Some Important Sampling Distribution120
7.4.1Chi-square Distribution120
7.4.2Student's Distribution(t-Distribution)124
Chapter 8Estimation and Uncertainty133
8.1Point Estimation133
8.1.1Some General Concepts of Point Estimation133
8.1.2Selection Criteria of Point Estimators135
8.2Method of Point Estimation142
8.2.1Method of Moments142
8.2.2Method of Maximum Likelihood144
8.3Interval Estimation149
8.3.1Basic Concepts of Confidence Intervals149
8.3.2Confidence Intervals for Parameters of a Normal Population151
8.3.3Confidence Intervals for the Difference of the Sample Means μ1-μ2155
8.4Confidence Interval for a Population Proportion p159
Chapter 9Hypothesis Testing164
9.1Basic Concepts and Principles of Hypothesis Testing164
9.1.1Hypothesis and Test Statistic164
9.1.2Errors in Hypothesis Testing167
9.2Hypotheses on a Single Normal Population168
9.2.1Hypothesis Concerning a Single Mean169
9.2.2Hypothesis Concerning a Single Variance171
9.3Two-Sample Tests of Hypotheses 174
9.3.1Tests on Two Means174
9.3.2Tests on Two Variances177
Chapter 10Application of R in Probability and Statistics182
10.1R Software Overview182
10.1.1Download and Installation of R Software182
10.1.2Using R as a Calculator183
10.1.3Defining and Using Variables184
10.1.5Plotting Graphs185
10.2R in Solving Probability and Statistical Problems187
10.2.1Probability Calculation187
10.2.2Plotting Statistical Graphs188
10.2.3Descriptive Statistics188
10.2.4Estimation in R190
10.2.5Testing Hypothesis on Mean and Variance of Normal Population195
Appendix Statistical Tables198
Table 1Poisson Distribution198
Table 2Standard Normal Distribution Function200
Table 3Values of χ2α201
Table 4Values of tα203
Table 5Values of Fα204
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