Information | Data |
---|---|
Instructor | Professor Marek Rychlik |
Office | Mathematics 605 |
Telephone | 1-520-621-6865 |
rychlik@arizona.edu | |
Instructor Homepage/Web Server | http://alamos.math.arizona.edu |
Course Homepage | http://alamos.math.arizona.edu/math466 |
Course Homepage (Mirror) | http://marekrychlik.com/math466 |
Personnel | Day of the Week | Hour | Room | Comment |
---|---|---|---|---|
Marek Rychlik | Monday | 4:00pm-5:00pm | Upper Division Tutoring via Teams (Zoom) | Upper Division Tutoring |
Marek Rychlik | Friday | 3:00pm-4:00pm | Math 466 Zoom Link | Regular office hours (Zoom, Math 466) |
Marek Rychlik | Friday | 4:00pm-5:00pm | Math 589 Zoom Link | Regular office hours (Zoom, Math 589) |
Office hours by appointment are welcome. Please contact me by e-mail first, so that I can activate a Zoom link for the meeting.
The course format is that of a conventional lecture, with in-class discussion and additional web-delivered content. All lectures will be recorded and available on Zoom and Panopto.
Written homework will be assigned regularly and graded using Gradescope.
In addition, the course incorporates required programming assignments. Numerical experimentation is essential to understanding and using the course subject matter. The assignments will be graded by an autograder implemented in Gradescope.
This course is an introduction to the theory of stochastics. The student will gain understanding of the following fundamental concepts:
In this course you are welcome and expected to use generative artificial intelligence/large language model tools, e.g. ChatGPT, Dall-e, Bard, Perplexity. Using these tools aligns with the course learning goals such as developing writing and programming skills, and ability to effectively use available information. Be aware that many AI companies collect information; do not enter confidential information as part of a prompt. LLMs may make up or hallucinate information. These tools may reflect misconceptions and biases of the data they were trained on and the human-written prompts used to steer them. You are responsible for checking facts, finding reliable sources for, and making a careful, critical examination of any work that you submit. Your use of AI tools or content must be acknowledged or cited. If you do not acknowledge or cite your use of an AI tool, what you submit will be considered a form of cheating or plagiarism. Please use the following guidelines for acknowledging/citing generative AI in your assignments:
Students are expected to be present for all exams. If a verifiable emergency arises which prevents you from taking an in-class exam at the regularly scheduled time, the instructor must be notified as soon as possible, and in any case, prior to the next regularly scheduled class. Make-up exams and quizzes will be administered only at the discretion of the instructor and only under extreme circumstances. If a student is allowed to make up a missed exam, (s)he must take it at a mutually arranged time. No further opportunities will be extended. Failure to contact your instructor as stated above or inability to produce sufficient evidence of a real emergency will result in a grade of zero on the exam. Other remedies, such as adjusting credit for other exams, may be considered.
Introduction to probability theory. Hoel, Paul G., Stone, Charles J., Port, Sidney C.
All examinations are planned to be administered during the class time, either in person or on Zoom.
If, due to unforseen circumstances, they cannot be held in person, they are held on Zoom using the "gallery view" mode.The exam papers for not in-person tests will be distributed on-line by D2L and collected electronically using D2L "dropbox" feature.
Exam or Assignment | Date | Grade contribution |
---|---|---|
Midterm 1 | February 15 (Thursday), 12:30pm - 1:45pm | 20% |
Midterm 2 | April 11 (Thursday), 12:30pm - 1:45pm | 20% |
Final Examination | May 8 (Wednesday), 1:00pm - 3:00pm | 30% |
Homework | See D2L | 30% |
Written homework consists of approximately twelve assignments equally contributing to the grade, each worth 30/12 = 2.5% of the grade. The assignments are posted on line at this link: Homework. The assignment papers are collected via Gradescope, which is cloud-based software for semi-automatic grading. Things to keep in mind:
Written homework is assigned regularly throughout the semester, for a total of approximately 80 problems. Two types of homework will be assigned:
Using Gradescope for grading differs from other grading systems. Mainly, it uses AI to allow the instructor to accurately grade a larger number of problems than it would be possible otherwise. Some grading is completely automated (e.g., solutions to problems with a numerical answer). More comples answers may be grouped automatically by using Machine Learning, OCR and image analysis. However, it is possible to completely confuse the system by improperly structuring the submitted document. Therefore, please read the instructions below carefully and re-visit them as needed. Note that Gradescope supports automatic regrade requests which you can use if all fails.
The solutions must be structured in such a way that Gradescope can read them and that its 'AI' can interpret them. Your homework must be submitted as a PDF document, even if you use scanner or phone to capture images. Two typical workflows will be as follows:
The class will have small programming assignments. It is expected that you will be using software to gain insights into the assigned problems and subject matter. The programming assignments must be submitted in formats supported by Gradescope and the instructor. The number of programming languages will be limited two two or three. R will be supported and it is encouraged that you use it as it is most compatible with the course content.
For illustrating some aspects of the course, I will be using these programs (easy to download and free to use):
The final examination is scheduled for: May 8 (Wednesday), 1:00pm - 3:00pm.
The time, data and general exam rules are set by the University and can be found at these links:
The student in the class normally receives a letter grade A, B, C, D or E.
The cut-offs for the grades are:
Grade | % Range |
---|---|
A | 90%+ |
B | 80-90% |
C | 70-80% |
D | 60-70% |
E | 0-60% |
Normally, individual tests and assignments will not be "curved". However, grade cut-offs may be lowered at the end of the semester (but not raised!) to reflect the difficulty of the assignments and other factors that may cause abnormal grade distribution.
The grade will be computed by D2L and the partial grade will be updated automatically by the system as soon as the individual grades are recorded.
General UA policy regarding grades and grading systems is available at https://catalog.arizona.edu/policy-type/grade-policiesFor a list of emergency procedures for all types of incidents, please visit the website of the Critical Incident Response Team (CIRT):
Our goal in this classroom is that learning experiences be as accessible as possible. If you anticipate or experience physical or academic barriers based on disability, please let me know immediately so that we can discuss options. You are also welcome to contact Disability Resources (520-621-3268) to establish reasonable accommodations. For additional information on Disability Resources and reasonable accommodations, please visit http://drc.arizona.edu/ .
If you have reasonable accommodations, please plan to meet with me by appointment or during office hours to discuss accommodations and how my course requirements and activities may impact your ability to fully participate. Please be aware that the accessible table and chairs in this room should remain available for students who find that standard classroom seating is not usable. Code of Academic Integrity Required language: Students are encouraged to share intellectual views and discuss freely the principles and applications of course materials. However, graded work/exercises must be the product of independent effort unless otherwise instructed. Students are expected to adhere to the UA Code of Academic Integrity as described in the UA General Catalog. See: http://deanofstudents.arizona.edu/academic-integrity/students/academic-integrity http://deanofstudents.arizona.edu/codeofacademicintegrity .
Date Event January 10, 2024 Classes Begin January 15, 2024 Martin Luther King Jr. Holiday - No Classes March 2 - 10, 2024 Spring Recess - No Classes May 1, 2024 Last Day of Classes and Laboratory Sessions May 2, 2024 Reading Day - No Classes or Finals May 3 - 9, 2024 Final Examinations May 10, 2024 Commencement May 10, 2024 Degree Award Date for Students Completing by Close of Spring Semester Spring 2024 - Undergraduate Regular Academic Session
Date Event 10/1/2023 Shopping Cart available 1/9/2024 Last day to file Undergraduate Leave of Absence 1/9/2024 Last day for students to add to or drop from a waitlist 1/10/2024 FIRST DAY OF SPRING CLASSES
UAccess still available for registration
First day to file for the Grade Replacement Opportunity (GRO)
First day to add classes for audit and instructor signature is required1/15/2024 Martin Luther King Day, no classes 1/17/2024 Last day to use UAccess for adding classes, changing classes, or changing sections 1/18/2024 Instructor approval required on a Change of Schedule form to ADD or CHANGE classes 1/23/2024 Last day to drop without a grade of W (withdraw)
Classes dropped on or before this date will remain on your UAccess academic record with a status of dropped, but will not appear on your transcript
Last day to change from credit to audit, or vice versa, with only an instructor's signature1/23/2024 Last day for a refund 1/24/2024 Beginning today, students may completely withdraw from all classes in the term 1/24/2024 W period begins a penalty grade of W will be awarded for each withdrawal and the class(es) will appear on your transcript
Beginning today, a change from credit to audit requires instructor approval on a Change of Schedule form2/1/2024 Last day to apply for Spring degree candidacy without a late fee After this date a $50 00 Late Candidacy Application fee will be assessed 2/6/2024 Last day for department staff to add or drop in UAccess 2/6/2024 Last day to change from pass/fail to regular grading or vice versa with only instructor approval on a Change of Schedule form 2/7/2024 Instructor's and dean's signatures are required on a Change of Schedule form to change from pass/fail to regular grades or vice versa 3/3/2024 Last day to make registration changes without the dean's signature 3/4/2024 Instructor's and dean's signatures are required on all Change of Schedule forms to ADD or CHANGE classes 3/4/2024 Spring recess begins 3/10/2024 Spring recess ends 3/26/2024 Last day to file for Grade Replacement Opportunity (GRO) 3/26/2024 Last day for students to withdraw from a class online through UAccess
Last day for students to change to/from audit with only instructor approval
Last day for instructors to administratively drop students3/27/2024 Instructor and dean's signatures required on a Late Change Petition in order to withdraw from class and students must have an extraordinary reason for approval
Beginning today, a change from credit to audit will be permitted only if the student is doing passing work on the course. Instructor and dean's permission required on a Change of Schedule form4/9/2024 Last day for students to submit a Late Change Petition to their college 5/1/2024 Last day to request a complete withdraw from all classes in the term 5/1/2024 Last day of class--no registration changes can be made after the last day of class 5/2/2024 Reading day, no classes 5/3/2024 Final exams begin 5/9/2024 Final exams end
Final grades are available in UAccess as soon as the instructor posts them
Per Faculty Senate Policy, grades should be submitted within two business days after the final exam5/10/2024 Degree award date
Chapter.Section | Title | Page | Covered | Date |
---|---|---|---|---|
What Is Statistics? | ||||
1.1 | Introduction | 1 | ||
1.2 | Characterizing a Set of Measurements: Graphical Methods | 3 | ||
1.3 | Characterizing a Set of Measurements: Numerical Methods | 8 | ||
1.4 | How Inferences Are Made | 13 | ||
1.5 | Theory and Reality | 14 | ||
1.6 | Summary | 15 | ||
Probability | ||||
2.1 | Introduction | 20 | ||
2.2 | Probability and Inference | 21 | ||
2.3 | A Review of Set Notation | 23 | ||
2.4 | A Probabilistic Model for an Experiment: The Discrete Case | 26 | ||
2.5 | Calculating the Probability of an Event: The Sample-Point Method | 35 | ||
2.6 | Tools for Counting Sample Points | 40 | ||
2.7 | Conditional Probability and the Independence of Events | 51 | ||
2.8 | Two Laws of Probability | 57 | ||
2.9 | Calculating the Probability of an Event: The Event-Composition Method | 62 | ||
2.10 | The Law of Total Probability and Bayes’ Rule | 70 | ||
2.11 | Numerical Events and Random Variables | 75 | ||
2.12 | Random Sampling | 77 | ||
2.13 | Summary | 79 | ||
Discrete Random Variables and Their Probability Distributions | ||||
3.1 | Basic Deﬁnition | 86 | ||
3.2 | The Probability Distribution for a Discrete Random Variable | 87 | ||
3.3 | The Expected Value of a Random Variable or a Function of a Random Variable | 91 | ||
3.4 | The Binomial Probability Distribution | 100 | ||
3.5 | The Geometric Probability Distribution | 114 | ||
3.6 | The Negative Binomial Probability Distribution (Optional) | 121 | ||
3.7 | The Hypergeometric Probability Distribution | 125 | ||
3.8 | The Poisson Probability Distribution | 131 | ||
3.9 | Moments and Moment-Generating Functions | 138 | ||
3.10 | Probability-Generating Functions (Optional) | 143 | ||
3.11 | Tchebysheff’s Theorem | 146 | ||
3.12 | Summary | 149 | ||
Continuous Variables and Their Probability Distributions | ||||
4.1 | Introduction | 157 | ||
4.2 | The Probability Distribution for a Continuous Random Variable | 158 | ||
4.3 | Expected Values for Continuous Random Variables | 170 | ||
4.4 | The Uniform Probability Distribution | 174 | ||
4.5 | The Normal Probability Distribution | 178 | ||
4.6 | The Gamma Probability Distribution | 185 | ||
4.7 | The Beta Probability Distribution | 194 | ||
4.8 | Some General Comments | 201 | ||
4.9 | Other Expected Values | 202 | ||
4.10 | Tchebysheff’s Theorem | 207 | ||
4.11 | Expectations of Discontinuous Functions and Mixed Probability Distributions (Optional) | 210 | ||
4.12 | Summary | 214 | ||
Multivariate Probability Distributions | ||||
5.1 | Introduction | 223 | ||
5.2 | Bivariate and Multivariate Probability Distributions | 224 | ||
5.3 | Marginal and Conditional Probability Distributions | 235 | ||
5.4 | Independent Random Variables | 247 | ||
5.5 | The Expected Value of a Function of Random Variables | 255 | ||
5.6 | Special Theorems | 258 | ||
5.7 | The Covariance of Two Random Variables | 264 | ||
5.8 | The Expected Value and Variance of Linear Functions of Random Variables | 270 | ||
5.9 | The Multinomial Probability Distribution | 279 | ||
5.10 | The Bivariate Normal Distribution (Optional) | 283 | ||
5.11 | Conditional Expectations | 285 | ||
5.12 | Summary | 290 | ||
Functions of Random Variables | ||||
6.1 | Introduction | 296 | ||
6.2 | Finding the Probability Distribution of a Function of Random Variables | 297 | ||
6.3 | The Method of Distribution Functions | 298 | ||
6.4 | The Method of Transformations | 310 | ||
6.5 | The Method of Moment-Generating Functions | 318 | ||
6.6 | Multivariable Transformations Using Jacobians (Optional) | 325 | ||
6.7 | Order Statistics | 333 | ||
6.8 | Summary | 341 | ||
Sampling Distributions and the Central Limit Theorem | ||||
7.1 | Introduction | 346 | ||
7.2 | Sampling Distributions Related to the Normal Distribution | 353 | ||
7.3 | The Central Limit Theorem | 370 | ||
7.4 | A Proof of the Central Limit Theorem (Optional) | 377 | ||
7.5 | The Normal Approximation to the Binomial Distribution | 378 | ||
7.6 | Summary | 385 | ||
Estimation | ||||
8.1 | Introduction | 390 | ||
8.2 | The Bias and Mean Square Error of Point Estimators | 392 | ||
8.3 | Some Common Unbiased Point Estimators | 396 | ||
8.4 | Evaluating the Goodness of a Point Estimator | 399 | ||
8.5 | Conﬁdence Intervals | 406 | ||
8.6 | Large-Sample Conﬁdence Intervals | 411 | ||
8.7 | Selecting the Sample Size | 421 | ||
8.8 | Small-Sample Conﬁdence Intervals for μ and μ1 − μ2 | 425 | ||
8.9 | Conﬁdence Intervals for σ^2 | 434 | ||
8.10 | Summary | 437 | ||
Properties of Point Estimators and Methods of Estimation | ||||
9.1 | Introduction | 444 | ||
9.2 | Relative Efﬁciency | 445 | ||
9.3 | Consistency | 448 | ||
9.4 | Sufﬁciency | 459 | ||
9.5 | The Rao–Blackwell Theorem and Minimum-Variance Unbiased Estimation | 464 | ||
9.6 | The Method of Moments | 472 | ||
9.7 | The Method of Maximum Likelihood | 476 | ||
9.8 | Some Large-Sample Properties of Maximum-Likelihood Estimators (Optional) | 483 | ||
9.9 | Summary | 485 | ||
Hypothesis Testing | ||||
10.1 | Introduction | 488 | ||
10.2 | Elements of a Statistical Test | 489 | ||
10.3 | Common Large-Sample Tests | 496 | ||
10.4 | Calculating Type II Error Probabilities and Finding the Sample Size for Z Tests | 507 | ||
10.5 | Relationships Between Hypothesis-Testing Procedures and Conﬁdence Intervals | 511 | ||
10.6 | Another Way to Report the Results of a Statistical Test: Attained Signiﬁcance Levels, or p-Values | 513 | ||
10.7 | Some Comments on the Theory of Hypothesis Testing | 518 | ||
10.8 | Small-Sample Hypothesis Testing for μ and μ1 − μ2 | 520 | ||
10.9 | Testing Hypotheses Concerning Variances | 530 | ||
10.10 | Power of Tests and the Neyman–Pearson Lemma | 540 | ||
10.11 | Likelihood Ratio Tests | 549 | ||
10.12 | Summary | 556 | ||
Linear Models and Estimation by Least Squares | ||||
11.1 | Introduction | 564 | ||
11.2 | Linear Statistical Models | 566 | ||
11.3 | The Method of Least Squares | 569 | ||
11.4 | Properties of the Least-Squares Estimators: Simple Linear Regression | 577 | ||
11.5 | Inferences Concerning the Parameters βi | 584 | ||
11.6 | Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression | 589 | ||
11.7 | Predicting a Particular Value of Y by Using Simple Linear Regression | 593 | ||
11.8 | Correlation | 598 | ||
11.9 | Some Practical Examples | 604 | ||
11.10 | Fitting the Linear Model by Using Matrices | 609 | ||
11.11 | Linear Functions of the Model Parameters: Multiple Linear Regression | 615 | ||
11.12 | Inferences Concerning Linear Functions of the Model Parameters: Multiple Linear Regression | 616 | ||
11.13 | Predicting a Particular Value of Y by Using Multiple Regression | 622 | ||
11.14 | A Test for H0 : βg+1 = βg+2 = · · · = βk = 0 | 624 | ||
11.15 | Summary and Concluding Remarks | 633 | ||
Considerations in Designing Experiments | ||||
12.1 | The Elements Affecting the Information in a Sample | 640 | ||
12.2 | Designing Experiments to Increase Accuracy | 641 | ||
12.3 | The Matched-Pairs Experiment | 644 | ||
12.4 | Some Elementary Experimental Designs | 651 | ||
12.5 | Summary | 657 | ||
The Analysis of Variance | ||||
13.1 | Introduction | 661 | ||
13.2 | The Analysis of Variance Procedure | 662 | ||
13.3 | Comparison of More Than Two Means: Analysis of Variance for a One-Way Layout | 667 | ||
13.4 | An Analysis of Variance Table for a One-Way Layout | 671 | ||
13.5 | A Statistical Model for the One-Way Layout | 677 | ||
13.6 | Proof of Additivity of the Sums of Squares and E(MST) for a One-Way Layout (Optional) | 679 | ||
13.7 | Estimation in the One-Way Layout | 681 | ||
13.8 | A Statistical Model for the Randomized Block Design | 686 | ||
13.9 | The Analysis of Variance for a Randomized Block Design | 688 | ||
13.10 | Estimation in the Randomized Block Design | 695 | ||
13.11 | Selecting the Sample Size | 696 | ||
13.12 | Simultaneous Conﬁdence Intervals for More Than One Parameter | 698 | ||
13.13 | Analysis of Variance Using Linear Models | 701 | ||
13.14 | Summary | 705 | ||
Analysis of Categorical Data | ||||
14.1 | A Description of the Experiment | 713 | ||
14.2 | The Chi-Square Test | 714 | ||
14.3 | A Test of a Hypothesis Concerning Speciﬁed Cell Probabilities: A Goodness-of-Fit Test | 716 | ||
14.4 | Contingency Tables | 721 | ||
14.5 | r × c Tables with Fixed Row or Column Totals | 729 | ||
14.6 | Other Applications | 734 | ||
14.7 | Summary and Concluding Remarks | 736 | ||
Nonparametric Statistics | ||||
15.1 | Introduction | 741 | ||
15.2 | A General Two-Sample Shift Model | 742 | ||
15.3 | The Sign Test for a Matched-Pairs Experiment | 744 | ||
15.4 | The Wilcoxon Signed-Rank Test for a Matched-Pairs Experiment | 750 | ||
15.5 | Using Ranks for Comparing Two Population Distributions: Independent Random Samples | 755 | ||
15.6 | The Mann–Whitney U Test: Independent Random Samples | 758 | ||
15.7 | The Kruskal–Wallis Test for the One-Way Layout | 765 | ||
15.8 | The Friedman Test for Randomized Block Designs | 771 | ||
15.9 | The Runs Test: A Test for Randomness | 777 | ||
15.10 | Rank Correlation Coefﬁcient | 783 | ||
15.11 | Some General Comments on Nonparametric Statistical Tests | 789 | ||
Introduction to Bayesian Methods for Inference | ||||
16.1 | Introduction | 796 | ||
16.2 | Bayesian Priors, Posteriors, and Estimators | 797 | ||
16.3 | Bayesian Credible Intervals | 808 | ||
16.4 | Bayesian Tests of Hypotheses | 813 | ||
16.5 | Summary and Additional Comments | 816 | ||
Appendix 1. | Matrices and Other Useful Mathematical Results | 821 | ||
A1.1 | Matrices and Matrix Algebra | 821 | ||
A1.2 | Addition of Matrices | 822 | ||
A1.3 | Multiplication of a Matrix by a Real Number | 823 | ||
A1.4 | Matrix Multiplication | 823 | ||
A1.5 | Identity Elements | 825 | ||
A1.6 | The Inverse of a Matrix | 827 | ||
A1.7 | The Transpose of a Matrix | 828 | ||
A1.8 | A Matrix Expression for a System of Simultaneous Linear Equations | 828 | ||
A1.9 | Inverting a Matrix | 830 | ||
A1.10 | Solving a System of Simultaneous Linear Equations | 834 | ||
A1.11 | Other Useful Mathematical Results | 835 | ||
Appendix 2. | Common Probability Distributions, Means, Variances, and Moment-Generating Functions | 837 | ||
Appendix 2.Table 1 | Discrete Distributions | 837 | ||
Appendix 2.Table 2 | Continuous Distributions | 838 | ||
Appendix 3. | Tables | 839 | ||
Appendix 3.Table 1 | Binomial Probabilities | 839 | ||
Appendix 3.Table 2 | Table of e−x | 842 | ||
Appendix 3.Table 3 | Poisson Probabilities | 843 | ||
Appendix 3.Table 4 | Normal Curve Areas | 848 | ||
Appendix 3.Table 5 | Percentage Points of the t Distributions | 849 | ||
Appendix 3.Table 6 | Percentage Points of the χ 2 Distributions | 850 | ||
Appendix 3.Table 7 | Percentage Points of the F Distributions | 852 | ||
Appendix 3.Table 8 | Distribution Function of U | 862 | ||
Appendix 3.Table 9 | Critical Values of T in the Wilcoxon Matched-Pairs, Signed-Ranks Test; n = 5(1)50 | 868 | ||
Appendix 3.Table 10 | Distribution of the Total Number of Runs R in Samples of Size (n 1 , n 2 ); P(R ≤ a) | 870 | ||
Appendix 3.Table 11 | Critical Values of Spearman’s Rank Correlation Coefﬁcient | 872 | ||
Appendix 3.Table 12 | Random Numbers | 873 | ||
Index | 896 |