{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–6. Sign up to view the full content.

Statistics 401: An Introduction to Statistics for Engineers and Scientists Michael G. Akritas Penn State University Fall 2006

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern