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401txt - Statistics 401 An Introduction to Statistics for...

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Statistics 401: An Introduction to Statistics for Engineers and Scientists Michael G. Akritas Penn State University Fall 2006
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Contents 1 Basic Statistical Concepts 1 1.1 Why Statistics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Populations and Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Random Variables and Statistical Populations . . . . . . . . . . . . . . . . 4 1.4 Population Average and Sample Average . . . . . . . . . . . . . . . . . . . 6 1.5 Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Some Sampling Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6.1 Representative Samples . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6.2 Simple Random Sampling, and Stratified Sampling . . . . . . . . . 11 1.6.3 Sampling With and Without Replacement . . . . . . . . . . . . . . 13 1.6.4 Non-representative Sampling . . . . . . . . . . . . . . . . . . . . . . 14 1.7 Statistical Experiments and Observational Studies . . . . . . . . . . . . . . 15 1.8 The Role of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.9 Approaches to Statistical Inference . . . . . . . . . . . . . . . . . . . . . . 18 1.10 Exercises for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Introduction to Probability 22 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Sample Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Events and Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 II
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2.4 Interpretation and Properties of Probability . . . . . . . . . . . . . . . . . 27 2.5 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 Counting Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.7 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7.1 The Law of Total Probability . . . . . . . . . . . . . . . . . . . . . 42 2.7.2 Bayes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.8 Exercises for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3 Random Variables and Their Distributions 50 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3.1 The cumulative distribution function . . . . . . . . . . . . . . . . . 54 3.3.2 The probability mass function of a discrete distribution . . . . . . . 57 3.3.3 The probability density function of a continuous distribution . . . . 60 3.4 Parameters of a Univariate Distribution . . . . . . . . . . . . . . . . . . . . 66 3.4.1 Discrete random variables . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.2 Continuous random variables . . . . . . . . . . . . . . . . . . . . . 73 3.5 Models for Discrete Random Variables . . . . . . . . . . . . . . . . . . . . 79 3.5.1 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . 80 3.5.2 The Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . 82 3.5.3 The Geometric and Negative Binomial Distributions . . . . . . . . 84 3.5.4 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . 85 3.6 Models for Continuous Random Variables . . . . . . . . . . . . . . . . . . 91 3.6.1 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . 92 3.6.2 Other Continuous Distributions . . . . . . . . . . . . . . . . . . . . 97 3.7 Exercises for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 III
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4 Multivariate Variables and Their Distribution 105 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Joint Distributions and the Joint CDF . . . . . . . . . . . . . . . . . . . . 106 4.3 The Joint Probability Mass Function . . . . . . . . . . . . . . . . . . . . . 109 4.3.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . 109 4.3.2 Marginal Probability Mass Functions . . . . . . . . . . . . . . . . . 110 4.3.3 Conditional Probability Mass Functions . . . . . . . . . . . . . . . 112 4.3.4 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.4 The Joint Probability Density Function . . . . . . . . . . . . . . . . . . . . 117 4.4.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . 117 4.4.2 Marginal Probability Density Functions . . . . . . . . . . . . . . . . 118 4.4.3 Conditional Probability Density Functions . . . . . . . . . . . . . . 119 4.4.4 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.5 Expected Value and Variance of a Statistic . . . . . . . . . . . . . . . . . . 122 4.5.1 Statistics and Sampling Distributions . . . . . . . . . . . . . . . . . 122 4.5.2 Expected Value of Sums . . . . . . . . . . . . . . . . . . . . . . . . 125 4.6 Parameters of a Multivariate Distribution . . . . . . . . . . . . . . . . . . 126 4.6.1 The Regression Function . . . . . . . . . . . . . . . . . . . . . . . . 126 4.6.2 Covariance and Correlation . . . . . . . . . . . . . . . . . . . . . . 129 4.7 Variance and Covariance of Sums . . . . . . . . . . . . . . . . . . . . . . . 134 4.8 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.9 Models for Joint Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.9.1 Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.9.2 Multinomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . 142 4.9.3 Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . 144 4.9.4 Distributions Derived from the Normal: χ 2 , t, and F . . . . . . . . 144 4.10 Exercises for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 IV
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5 Descriptive Statistics 157 5.1 Graphical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.1.1 Stem-and-Leaf Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 158 5.1.2 Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . 160 5.1.3 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.1.4 Bar Graphs for Qualitative Data . . . . . . . . . . . . . . . . . . . 162 5.1.5 The Empirical Distribution Function . . . . . . . . . . . . . . . . . 162 5.1.6 P-P and Q-Q plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.1.7 Dot Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.2 Numerical Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.2.1 Measures of Location . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.2.2
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