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Introduction to Probability Theory
Purpose of Course showclose
This course will introduce you to the fundamentals of probability theory and random processes. The theory of probability was originally developed in the 17th century by two great French mathematicians, Blaise Pascal and Pierre de Fermat, to understand gambling. Today, the theory of probability has found many applications in science and engineering. In this course, you will learn the basic terminology and concepts of probability theory, including sample size, random experiments, outcome spaces, discrete distribution, probability density function, expected values, and conditional probability. You will also learn about the fundamental properties of several special distributions, including binomial, geometric, normal, exponential, and Poisson distributions.
Course Information showclose
Course Designer: Tuan Dinh
Primary Resources: This course is comprised of a range of different free, online materials. However, the course makes primary use of the following:
- MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics
- UCLA Courses: Professor Herbert Enderton’s Mathematics 3C Video Lectures (YouTube)
- Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability
- Khan Academy: Salman Khan’s Lecture Series on Probability
In order to complete this course, you will need to earn a 70% or higher on the Final Exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.
Time Commitment: The materials for this course will take approximately 104 hours to complete.
Tips/Suggestions: You may not understand everything in the readings the first time around. Do not get frustrated. For each subunit, you may want to start with the readings from Professors Charles M. Grinstead and J. Laurie Snell’s book, Introduction to Probability, and the video lectures first. Use Professor Dmitry Panchenko’s lecture notes as the summary of the key points of each subunit. Complete all homework and quizzes in each unit, which will help you apply what you just learned.
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This course features a number of Khan Academy™ videos. Khan Academy™ has a library of over 3,000 videos covering a range of topics (math, physics, chemistry, finance, history and more), plus over 300 practice exercises. All Khan Academy™ materials are available for free at www.khanacademy.org.
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Learning Outcomes showclose
- Define probability, outcome space, events, and probability functions.
- Use combinations to evaluate the probability of outcomes in coin-flipping experiments.
- Calculate the union of events and conditional probability.
- Apply Bayes’s theorem to simple situations.
- Calculate the expected values of discrete and continuous distributions.
- Calculate the sums of random variables.
- Calculate cumulative distributions and marginal distributions.
- Evaluate random processes governed by binomial, multinomial, geometric, exponential, normal, and Poisson distributions.
- Define the law of large numbers and the central limit theorem.
Course Requirements showclose
√ Have access to a computer.
√ Have continuous broadband Internet access.
√ Have the ability/permission to install plug-ins or software (e.g., Adobe Reader or Flash).
√ Have the ability to download and save files and documents to a computer.
√ Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).
√ Be competent in the English language.
√ Have read the Saylor Student Handbook.
√ Have completed MA101, MA102, MA103, MA211, and MA221, or their equivalents.
Unit Outline show close
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Unit 1: Introduction to Probability
This unit will introduce you to fundamental concepts of probability theory. You will learn the definitions of probability, random variables, outcome space, events, and probability function through simple chance experiments with discrete outcomes, such as tossing a coin or rolling a die. Random variables represent outcomes of chance experiments. You will also learn about four basic set operations—namely, union, intersection, difference, and complement. This unit will also introduce you to the concepts of conditional probability (i.e., the probability of event A, given the occurrence of some other event B) and Bayes’s theorem, which is one of the most celebrated theorems in the theory of probability.
Unit 1 Time Advisory show close
Unit 1 Learning Outcomes show close
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1.1 Probability and Set Operations
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 1: Probability, Set Operations”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 1: Probability, Set Operations” (PDF).
Instructions: Please read the PDF in its entirety. This reading will introduce you to the basic concepts of probability.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Reading: UC San Diego: Professor David A. Meyer’s Introduction to Probability: “Lecture 1”
Link: UC San Diego: Professor David A. Meyer’s Introduction to Probability: “Lecture 1” (PDF)
Instructions: Please click the link for to go to Professor Meyer’s Math 180A course page. Scroll down to where it says 7 Jan 08, and click on the “lecture notes” link. This reading will introduce you to the basic concepts of probability.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 1: Introduction to Probability and Counting”
Link: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 1: Introduction to Probability and Counting” (YouTube)
Instructions: Please watch this video (approximately 44 minutes), which will introduce you to the basic concepts of probability, including outcome space, events, and probability functions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Web Media: Khan Academy: Salman Khan’s Lecture Series on Probability: “Probability (1)” and “Probability (2)”
Link: Khan Academy: Salman Khan’s Lecture Series on Probability: “Probability (1)” (YouTube) and “Probability (2)” (YouTube)
Also available in:
iTunes U (1)
iTunes U (2)
Instructions: Please watch this series of videos. The first (approximately 11 minutes) defines probability. The second (approximately 10 minutes) explores the outcomes from coin flips.
Terms of Use: These videos are licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. They are attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 1: Probability, Set Operations”
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1.2 Properties of Probability and Combinations
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 2: Properties of Probability”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 2: Properties of Probability” (PDF)
Instructions: Please the read the PDF in its entirety. This reading will introduce you to the basic properties of probability.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Web Media: Khan Academy: Salman Khan’s Lecture Series on Probability: “Probability Using Combinations”
Link: Khan Academy: Salman Khan’s Lecture Series on Probability: “Probability Using Combinations” (YouTube)
Also available in:
iTunes U
Instructions: Please watch this video (approximately 9 minutes). This video will help you learn how to calculate probability from coin flips by using combinations.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 2: Properties of Probability”
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1.3 Union of Events
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 3: Union of Events”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 3: Multinomial Coefficients, Union of Events” (PDF)
Instructions: Please read the PDF in its entirety. This reading will help you learn how to calculate the union of events.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 3: Union of Events”
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1.4 Conditional Probability
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 4: Matching Problem, Conditional Probability”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 4: Matching Problem, Conditional Probability” (PDF)
Instructions: Please read the PDF in its entirety. This reading will help you learn about the definition of conditional probability. Calculations of conditional probability are illustrated via a simple example.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Web Media: Khan Academy: Salman Khan’s Lecture Series on Probability: “Conditional Probability and Combinations”
Link: Khan Academy: Salman Khan’s Lecture Series on Probability: “Conditional Probability and Combinations” (YouTube)
Also available in:
iTunes U
Instructions: Please watch this video (approximately 17 minutes). This video will help you learn how to calculate the conditional probability of using a fair coin given that one flipped 4 out of 6 heads.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 4: Matching Problem, Conditional Probability”
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1.5 Independent events
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 5: Independence of Events”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 5: Independence of Events” (PDF)
Instructions: Please read the PDF in its entirety. This reading will introduce you to one of the most important formulas in probability theory. This reading will also demonstrate the power of Bayes’s theorem through several real-world examples.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Lecture: UCLA: Professor David Welsbart’s Mathematics 3C: “Lecture 7: Independent Events”
Link: UCLA: Professor David Welsbart’s Mathematics 3C: “Lecture 7: Independent Events” (YouTube)
Instructions: Please watch this video (approximately 39 minutes minutes), which will introduce you to independent events.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Assessment: The Saylor Foundation’s “Unit 1.5 Quiz”
Link: The Saylor Foundation’s “Unit 1.5 Quiz” (PDF)
Instructions: Please complete the linked assessment. When you are done, please check your work against The Saylor Foundation’s "Answer Key" for subunit 1.5 (PDF). This assessment should take you no longer than 1 hour to complete.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 5: Independence of Events”
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1.6 Bayes’s Theorem
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 7: Bayes’ Formula”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 7: Bayes’ Formula” (PDF)
Instructions: Please read the PDF in its entirety. This reading will introduce you to one of the most important formulas in probability theory. This reading will also demonstrate the power of Bayes’s theorem through several real-world examples.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Web Media: Khan Academy: Salman Khan’s Lecture Series on Probability: “Probability (8)”
Link: Khan Academy: Salman Khan’s Lecture Series on Probability: “Probability (8)” (YouTube)
Also available in:
iTunes U
Instructions: Please watch this video (approximately 5 minutes). This video will introduce you to Bayes’s theorem.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Assessment: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Practice Test 1”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Practice Test 1” (PDF)
Instructions: Please click the link to open “Practice Test 1.” Solve problems 1 and 2. Read the problems carefully and then try to solve them yourself before looking at the solutions.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 7: Bayes’ Formula”
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Unit 1 Assessment
- Assessment: The Saylor Foundation’s “Unit 1 Assessment”
Link: The Saylor Foundation’s “Unit 1 Assessment” (PDF) and “Unit 1 Answer Key” (PDF).
Instructions: Please click on the link to open the multiple choice quiz for Unit 1. Write down your choices and compare with the Answer Key after you answer all questions.See a broken link? Please let us know!
- Assessment: The Saylor Foundation's "Unit 1 Multiple Choice Quiz"
Link: The Saylor Foundation's "Unit 1 Assessment Multiple Choice Quiz" (PDF) and "Unit 1 Assessment Multiple Choice Quiz Answer Key" (PDF).
Instructions: Please answer the multiple choice questions and then check your answers using the answer key.See a broken link? Please let us know!
- Assessment: The Saylor Foundation’s “Unit 1 Assessment”
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Unit 2: Introduction to Probability Distributions
This unit will introduce you to probability distributions and several of their most important properties. You will learn how to identify discrete and continuous probability distributions as well as calculate their expected values and variances. You will also learn how to calculate the cumulative probability distribution for a given probability distribution and vice versa.
Unit 2 Time Advisory show close
Unit 2 Learning Outcomes show close
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2.1 Random Variables and Distributions
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 8: Random Variables and Distributions”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 8: Random Variables and Distributions” (PDF)
Instructions: Please read the PDF in its entirety. This reading will help you learn about random variables, which map probabilities to real numbers. This reading will also introduce you to discrete and continuous distributions as well as several examples of discrete and continuous distributions.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Reading: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 1: Discrete Probability Distributions” and “Chapter 2: Continuous Probability Densities”
Link: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability (PDF):“Chapter 1: Discrete Probability Distributions” and “Chapter 2: Continuous Probability Densities”
Instructions: The book is in PDF format. Save a copy of the book for future use. For this unit, please read “Chapter 1: Discrete Probability Distributions” (pages 18–24). This reading will introduce you to random variables and probability distributions. You will also learn about several important properties of probability distributions.
Terms of Use: Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find the original version here.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 8: Random Variables and Distributions”
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2.2 Expected Values
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 9: Expected Values”
Link: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 9: Expected Values” (YouTube)
Instructions: Please watch this video (approximately 50 minutes), which will introduce you to how to calculate expected values of a probability distribution.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Reading: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 6: Expected Value and Variance”
Link: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J.Laurie Snell’s Introduction to Probability: (PDF)“Chapter 6: Expected Value and Variance”
Instructions: The book is in PDF format. Save a copy of the book for future use. For this unit, please read sections 6.1 to 6.3 of “Chapter 6: Expected Value and Variance” (pages 225–278). In this reading, you will learn how to evaluate sums of random discrete and continuous variables.
Terms of Use: Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find the original version here.See a broken link? Please let us know!
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 9: Expected Values”
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2.3 Sum of Random Variables
- Reading: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 7: Sum of Random Variables”
Link: Dartmouth College and Swarthmore College: Professors Charles M.Grinstead and J.Laurie Snell’s Introduction to Probability: (PDF) “Chapter 7: Sum of Random Variables”
Instructions: The book is in PDF format. Save a copy of the book for future use. For this unit, please read sections 7.1 and 7.2 of “Chapter 7: Sum of Random Variables” (pages 285–300). This reading will help you learn how to evaluate sums of random discrete and continuous variables.
Terms of Use: Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find the original version here.See a broken link? Please let us know!
- Reading: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 7: Sum of Random Variables”
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2.4 Cumulative Distribution
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 9: Cumulative Distribution Function”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 9: Cumulative Distribution Function” (PDF)
Instructions: Please read the PDF in its entirety. This reading will help you learn how to calculate cumulative distributions for discrete and continuous probability distributions.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Assessment: The Saylor Foundation’s “Unit 2 Assessment”
Link: The Saylor Foundation’s “Unit 2 Assessment” (PDF)
Instructions: Please complete the linked assessment. When you are done, please check your work against The Saylor Foundation’s “Answer Key" for subunit 2.4 (PDF). This assessment should take you no longer than 1.5 hours to complete.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 9: Cumulative Distribution Function”
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2.5 Marginal Distribution
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 10: Marginal Distributions”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 10: Marginal Distributions” (PDF)
Instructions: Please click the link to for “Lecture 10: Marginal Distributions.” This reading will introduce you to marginal distributions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 10: Marginal Distributions”
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Unit 2 Assessment
- Assessment: The Saylor Foundation's "Unit 2 Assessment Multiple Choice Quiz"
Link: The Saylor Foundation's "Unit 2 Assessment Multiple Choice Quiz" (PDF) and "Unit 2 Assessment Multiple Choice Quiz Answer Key" (PDF)
Instructions: Please answer the multiple choice questions and then check your answers using the answer key.See a broken link? Please let us know!
- Assessment: The Saylor Foundation's "Unit 2 Assessment Multiple Choice Quiz"
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Unit 3: Discrete Distributions
In this unit, you will learn about four basic discrete probability distributions that have widespread applications in engineering and science: binomial distributions, multinomial distributions, geometric distributions, and Poisson distributions.
Unit 3 Time Advisory show close
Unit 3 Learning Outcomes show close
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3.1 Binomial Distributions
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 10: Binomial Distributions”
Link: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 10: Binomial Distributions” (YouTube)
Instructions: Please watch this video (approximately 51 minutes), which will introduce you to one of the most common discrete probability distributions: binomial distributions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Reading: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 5: Distributions and Densities”
Link: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: (PDF) “Chapter 5: Distributions and Densities”
Instructions: The book is in PDF format. Save a copy of the book for future use. For this unit, please read page 184 (the “Binomial Distribution” section) in “Chapter 5: Distributions and Densities.” This reading will introduce you to the definition of binomial distributions.
Terms of Use: Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find the original version here.See a broken link? Please let us know!
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 10: Binomial Distributions”
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3.2 Multinomial Distributions
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 12: Multinomial Distributions”
Link: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 12: Multinomial Distributions” (YouTube)
Instructions: Please watch this video (approximately 41 minutes), which will introduce you to multinomial distributions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 12: Multinomial Distributions”
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3.3 Geometric Distributions
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 13: Geometric Distributions”
Link: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 13: Geometric Distributions” (YouTube)
Instructions: Please watch this video (approximately 44 minutes), which will introduce you to multinomial distributions and geometric distributions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Reading: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 5: Distributions and Densities”
Link: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: (PDF) “Chapter 5: Distributions and Densities”
Instructions: The book is in PDF format. Save a copy of the book for future use. For this unit, please read pages 184–186 (the “Geometric Distributions” section) in “Chapter 5: Distributions and Densities.” This reading will help you learn about the definition of geometrical distributions.
Terms of Use: Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find the original version here.See a broken link? Please let us know!
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 13: Geometric Distributions”
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3.4 Poisson Distributions
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 14: Poisson Distribution” and “Lecture 15: Poisson Distribution (continued)”
Link: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 14: Poisson Distribution” (YouTube) and “Lecture 15: Poisson Distribution (continued)” (YouTube)
Instructions: Please watch these video lectures (approximately 47 minutes and 51 minutes, respectively), which will introduce you to Poisson distributions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 20: Poisson Distribution, Approximation of Binomial Distribution, Normal Distribution”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 20: Poisson Distribution, Approximation of Binomial Distribution, Normal Distribution” (PDF)
Instructions: Please read the PDF in its entirety. This reading will help you learn about Poisson distributions and how to approximate them by using binomial and normal distributions.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Reading: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 5: Distributions and Densities”
Link: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: (PDF) “Chapter 5: Distributions and Densities”
Instructions: The book is in PDF format. Save a copy of the book for future use. For this unit, please read pages 187–192 (the “Poisson Distribution” section) in “Chapter 5: Distributions and Densities.” This reading will help you learn about the definition of Poisson distributions.
Terms of Use: Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find the original version here.See a broken link? Please let us know!
- Assessment: The Saylor Foundation’s “Unit 3.4 Assessment”
Link: The Saylor Foundation’s “Unit 3.4 Assessment” (PDF)
Instructions: Please complete the linked assessment. When you are done, please check your work against The Saylor Foundation’s "Answer Key" for subunit 3.4 (PDF). This assessment should take you no longer than 1.5 hours to complete.See a broken link? Please let us know!
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 14: Poisson Distribution” and “Lecture 15: Poisson Distribution (continued)”
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Unit 3 Assessment
- Assessment: The Saylor Foundation's "Unit 3 Assessment Multiple Choice Quiz"
Link: The Saylor Foundation's "Unit 3 Assessment Multiple Choice Quiz" (PDF) and "Unit 3 Assessment Multiple Choice Quiz Answer Key" (PDF).
Instructions: Please answer the multiple choice questions and then check your answers using the answer key.See a broken link? Please let us know!
- Assessment: The Saylor Foundation's "Unit 3 Assessment Multiple Choice Quiz"
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Unit 4: Continuous Distributions
This unit will introduce to several important continuous probability distributions, including exponential distributions and normal distributions. You will also learn about standard normal distributions (i.e., a normal distribution with a mean of 0 and a standard deviation of 1). Normal distributions can be converted to standard normal distributions by using a standard formula.
Unit 4 Time Advisory show close
Unit 4 Learning Outcomes show close
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4.1 Probability Density Functions
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 16: Density Functions”
Link: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 16: Density Functions” (YouTube)
Instructions: Please watch this video (approximately 50 minutes), which will introduce you to probability density functions of continuous random variables.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 16: Density Functions”
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4.2 Exponential Distributions
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 17: Exponential Distributions”
Link: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 17: Exponential Distributions” (YouTube)
Instructions: Please watch this video (approximately 51 minutes), which will introduce you to common properties of exponential distributions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 17: Exponential Distributions”
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4.3 Normal Distributions
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 18: Normal Distribution”; and “Lecture 19: Normal Distribution (continued)”
Link: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 18: Normal Distribution” (YouTube) and “Lecture 19: Normal Distribution (continued)” (YouTube)
Instructions: Please watch these video lectures (approximately 49 minutes, and 50 minutes, respectively), which will introduce you to normal distributions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 21: Normal Distribution, Central Limit Theorem”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 21: Normal Distribution, Central Limit Theorem” (PDF)
Instructions: Please read the PDF in its entirety. This reading will help you learn about normal distributions.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Assessment: The Saylor Foundation’s “Unit 4.3 Assessment”
Link: The Saylor Foundation’s “Unit 4.3 Assessment” (PDF)
Instructions: Please complete the linked assessment. When you are done, please check your work against The Saylor Foundation’s "Answer Key" for subunit 4.3 (PDF). This assessment should take you no longer than 2 hours to complete.See a broken link? Please let us know!
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 18: Normal Distribution”; and “Lecture 19: Normal Distribution (continued)”
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4.4 Standard Normal Distributions
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 20: Standard Normal Distributions”
Link: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 20: Standard Normal Distributions” (YouTube)
Instructions: Please watch these video lectures (approximately 48 minutes), which will introduce you to standard normal distributions and transformations between standard normal distributions and other normal distributions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Assessment: University of Dayton: Professor Joe Mashburn’s MTH 114: “Homework 16”
Link: University of Dayton: Professor Joe Mashburn’s MTH 114: “Homework 16” (PDF)
Instructions: Please click the link for “Homework 16.” Solve problems 1 to 3. Read the problems carefully and then try to solve them yourself before looking up the solutions. When you are done, check your work against the solutions to “Homework 16” on the same webpage.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 20: Standard Normal Distributions”
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Unit 5: Law of Large Numbers and Central Limit Theorem
In this unit, you will learn about two important theorems of probability theory: the law of large numbers and the central limit theorem. The law of large numbers describes the result of performing the same experiment a large number of times. The central limit theorem states the conditions under which the mean of a large number of random variables will be normally distributed.
Unit 5 Time Advisory show close
Unit 5 Learning Outcomes show close
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5.1 Law of Large Numbers
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 18: Law of Large Numbers, Median”
Link: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 18: Law of Large Numbers, Median” (PDF)
Instructions: Please read pages 53–54 from the PDF. This reading will introduce you to the law of large numbers.
Terms of Use: Dmitry Panchenko. Introduction to Probability and Statistics, Spring, 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 5/19/2011). License: Creative Commons BY-NC-SA 3.0. You can find the original version here.See a broken link? Please let us know!
- Reading: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 8: Law of Large Numbers”
Link: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: (PDF) “Chapter 8: Law of Large Numbers”
Instructions: The book is in PDF format. Save a copy of the book for future use. For this unit, please read sections 8.1 and 8.2 of “Chapter 8: Law of Large Numbers” (pages 305–320). This reading discusses the law of large numbers.
Terms of Use: Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find the original version here.See a broken link? Please let us know!
- Reading: MIT OpenCourseWare: Professor Dmitry Panchenko’s Introduction to Probability and Statistics: “Lecture 18: Law of Large Numbers, Median”
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5.2 Central Limit Theorem
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 21: Central Limit Theorem”
Link: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 21: Central Limit Theorem” (YouTube)
Instructions: Please watch this video (approximately 53 minutes), which will introduce you to central limit theorem.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Reading: Dartmouth College and Swarthmore College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 9: Central Limit Theorem”
Link: Dartmouth College and Swarthmore College: Professors Charles M.Grinstead and J. Laurie Snell’s Introduction to Probability: (PDF) “Chapter 9: Central Limit Theorem”
Instructions: The book is in PDF format. Save a copy of the book for future use. For this unit, please read sections 9.1 to 9.3 of “Chapter 9: Central Limit Theorem” (pages 325–361). This reading discusses the second fundamental theorem of probability: the central limit theorem.
Terms of Use: Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find the original version here.See a broken link? Please let us know!
- Assessment: The Saylor Foundation’s “Unit 5.2 Assessment”
Link: The Saylor Foundation’s “Unit 5.2 Assessment” (PDF)
Instructions: Please complete the linked assessment. When you are done, please check your work against The Saylor Foundation’s "Answer Key" for subunit 5.2 (PDF). This assessment should take you no longer than 3 hours to complete.See a broken link? Please let us know!
- Lecture: UCLA: Professor Herbert Enderton’s Mathematics 3C: “Lecture 21: Central Limit Theorem”
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Final Exam
- Final Exam: The Saylor Foundation's MA252 Final Exam
Link: The Saylor Foundation's MA252 Final Exam
Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.See a broken link? Please let us know!
- Final Exam: The Saylor Foundation's MA252 Final Exam
