A First Course in Probability Models and Statistical Inference

1 — Introduction to Probability Models of the Real World.- 1.1 Probability Distributions of Random Variables.- Probability Models.- Random Variables and Their Random Experiments.- 1.2 Parameters to Characterize a Probability Distribution.- The Expected Value or Mean of a Random Variable.- The Variance, Measuring the Accuracy of the Mean.- 1.3 Linear Functions of a Random Variable.- 1.4 The Fundamentals of Probability Theory.- Three Basic Rules of Probability.- Bayes’ Theorem.- Random Experiments with Equally Likely Outcomes.- Chebyshev’s Theorem.- 1.5 Some Review Exercises.- 2 —Understanding Observed Data.- 2.1 Observed Data from the Real World.- Presenting Data Graphically.- Collecting Data.- Simple Random Samples Drawn from a Probability Distribution.- Populations.- Statistical Questions.- Simple Random Samples Drawn from a Population.- 2.2 Presenting and Summarizing Observed Numeric Data.- Measures of Centrality for Observed Numeric Data.- Measures of Spread for Observed Numeric Data.- 2.3 Grouped Data: Suppressing Irrelevant Detail.- Grouped Distributions of Observed Real World Data.- Histograms: Graphical Display of Grouped Relative Frequency Distributions.- 2.4 Using the Computer.- Describing, Picturing, and Comparing Population and Sample Data.- 3 — Discrete Probability Models.- 3.1 Introduction.- 3.2 The Discrete Uniform Distribution.- 3.3 The Hypergeometric Distribution.- Counting Rules.- What Is the Hypergeometric Model.- Calculating the Probabilities.- The Formulas.- 3.4 Sampling with Replacement from a Dichotomous Population.- What Is the Model?.- The Formulas.- 3.5 The Bernoulli Trial.- 3.6 The Geometric Distribution.- What Is the Model?.- The Formulas.- 3.7 The Binomial Distribution.- The Binomial Experiment.- The Binomial Random Variable Itself.- 3.8 The Poisson Distribution.- 3.9 The Negative Binomial Distribution.- 3.10 Some Review Problems.- 4 — Continuous Probability Models.- 4.1 Continuous Distributions and the Continuous Uniform Distribution.- Continuous Distributions.- The Probability Density function.- The Continuous Uniform Distribution.- 4.2 The Exponential Distribution.- Modeling the Reliability of a System.- The Exponential Distribution.- 4.3 The Normal Distribution.- The Normal Distribution as a Model for Measurement Error.- The Normal Distribution as an Abstract Model.- The Standardizing Transformation.- The Normal Probability Plot.- Continuous Approximations to Integer-Valued Random Variables.- The Normal Approximation to the Binomial.- 4.4 The Chi-Squared Distribution.- 4.5 A Few Review Problems.- 5 — Estimation of Parameters.- 5.1 Parameters and Their Estimators.- Estimators—the Entire Context.- 5.2 Estimating an Unknown Proportion.- The Sampling Distribution for % MathType!MTEF!2!1!+-
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$$.- Too Many Approximations: Are They Valid?.- The Appropriate Sample Size for a Given Error Tolerance.- 5.3 Estimating an Unknown Mean.- The Estimator % MathType!MTEF!2!1!+-
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$$, Using s Instead of ? Takes Us to Student’s t-Distribution When the Distribution You’re Sampling from Is Normal.- 5.4 A Confidence Interval Estimate for an Unknown ?.- 5.5 One-Sided Intervals, Prediction Intervals, Tolerance Intervals.- One-Sided Confidence Intervals.- Prediction Intervals for Observations from a Normal Distribution.- Tolerance Intervals.- 6 — Introduction to Tests of Statistical Hypotheses.- 6.1 Introduction.- Statistical Hypotheses.- What Are These Two Testing Procedures?.- Contrasting the Two Testing Procedures.- 6.2 Tests of Significance.- A Dialogue.- The p-Value.- Comparing Means and Comparing Proportions (Large Samples): Two New Parameters and Their Estimators.- Practical Versus Statistical Significance.- The Test of Significance as an Argument by Contradiction.- What If the p-Value is Not Small?.- A Case Where “Not Small p-Value” Is Conclusive and “Small” Not.- Chi-Squared Tests for Goodness of Fit, Homogeneity, and Independence.- 6.3 Hypothesis Tests.- Setting Up the Hypothesis Test.- The Possible Errors.- Real-World Interpretation of the Conclusions and Errors.- Moving in the Direction of Common Practice.- The Rejection Region.- The Decision Rule and Test Statistic p-Values for Hypothesis Tests 273.- Controlling Power and Type II Error.- 6.4 A Somewhat Comprehensive Review.- 7 — Introduction to Simple Linear Regression.- 7.1 The Simple Linear Regression Model.- 7.2 The Least Squares Estimates for ? and ?.- The Principle of Least Squares.- Calculating the Least Squares Estimate of ? and ?.- 7.3 Using the Simple Linear Regression Model.- Testing Hypotheses Concerning ?.- The Coefficient of Determination.- Confidence Intervals to Predict ?Y|X or the Average of a Few Y’s for Xp, a Particular Value of X.- 7.4 Some Review Problems.- The Data.- Answers to Try Your Hand—Level 1.- Answers to Try Your Hand—Level II.- Tables.- The Standard Normal Distribution.- The Chi-Squared Distribution.- Index of Notation.- Author Index.