**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**

Completing this unit should take approximately 22 hours.

☐ Subunit 5.1: 8.5 hours

☐ Subunit 5.2: 8 hours

☐ Subunit 5.3: 5.5 hours

**Unit5 Learning Outcomes**

Upon successful completion of this unit, you will be able to:
- use the law of large numbers to evaluate samples of random
variables; and

- use the central limit theorem to estimate distributions of sample
means.

**5.1 Law of Large Numbers**
- **Reading: Dartmouth College: Professors Charles M. Grinstead and J.
Laurie Snell’s Introduction to Probability: “Chapter 8, Section
8.1: Law of Large Numbers for Discrete Random Variables” and
“Chapter 8, Section 8.2: Law of Large Numbers for Continuous Random
Variables”**
Link: Dartmouth College: Professors Charles M. Grinstead and J.
Laurie Snell’s

*Introduction to Probability*: “Chapter 8, Section 8.1: Law of Large Numbers for Discrete Random Variables” (PDF) and “Chapter 8, Section 8.2: Law of Large Numbers for Continuous Random Variables” (PDF)

Instructions: Read Section 8.1 and Section 8.2 of “Chapter 8: Law of Large Numbers” on pages 305 - 320. This reading discusses the law of large numbers. If you have n independent, identically distributed random variables X1, X2, ..., Xn, let S be their sum. The law of large numbers says roughly that if n is very large, then S/n is a good approximation of the mean E(Xn). It is given as a limit theorem. Some nice applications will be given.

Reading these textbook sections and taking notes should take approximately 4 hours.

Terms of Use: This resource is licensed under a GNU Free Documentation License (FDL). It is attributed to Charles M. Grinstead, J. Laurie Snell and Dartmouth College.

**Reading: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s**Link: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s*Math 18.05: Introduction to Probability and Statistics*: “Lecture 18: Law of Large Numbers, Median”*Math 18.05: Introduction to Probability and Statistics*: “Lecture 18: Law of Large Numbers, Median” (PDF)

Instructions: Read the section of the law of large numbers on pages 53 - 54 of this lecture. This reading will further your understanding of the law of large numbers.

Reading this lecture and taking notes should take approximately 2 hours.

Terms of Use: The above material is released under a Creative Commons Attribution-Non Commercial-ShareAlike License 3.0. It is attributed to Dmitry Panchenko and Massachusetts Institute of Technology OpenCourseWare.**Assessment: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s**Link: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s*Introduction to Probability*: “Chapter 8, Section 8.1: Exercises” and “Chapter 8, Section 8.2: Exercises”*Introduction to Probability*: “Chapter 8, Section 8.1: Exercises” (PDF) and “Chapter 8, Section 8.2: Exercises” (PDF)

Instructions: Go to the Section 8.1 exercises on pages 312 - 313 and complete exercises 1, 5, and 11. Then go to the Section 8.2 exercises on pages 321 - 322 and complete exercises 1, 2, 4, and 10. You can then check your answers to odd-numbered questions here.

Completing this assessment should take approximately 2 hours and 30 minutes.

Terms of Use: This resource is licensed under a GNU Free Documentation License (FDL). It is attributed to Charles M. Grinstead, J. Laurie Snell and Dartmouth College.

**5.2 Central Limit Theorem**
- **Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C:
Math and Probability for Life Sciences: “Lecture 21: Central Limit
Theorem”**
Link: YouTube: UCLA: Professor Herbert Enderton’s

*Math 3C: Math and Probability for Life Sciences*: “Lecture 21: Central Limit Theorem” (YouTube)

Instructions: Watch this video for an introduction to the Central Limit Theorem. The central limit theorem roughly says that if you select samples of size n randomly from a population of X values, where X is a random variable that has any distribution, then as n gets very large, the sample means will approach a normal distribution. Formulas for the mean and standard deviation of that normal distribution will be given. This is one of the most important concepts in statistics.

Watching this video and taking notes should take approximately 1 hour and 30 minutes.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to UCLA and Dr. Herbert Enderton, and the original version can be found here.

**Reading: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s**Link: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s*Introduction to Probability*: “Chapter 9, Section 9.1: Central Limit Theorem for Bernoulli Trials,” “Chapter 9, Section 9.2: Central Limit Theorem for Discrete Independent Trials,” “Chapter 9, Section 9.3: Central Limit Theorem for Continuous Independent Trials”*Introduction to Probability*: “Chapter 9, Section 9.1: Central Limit Theorem for Bernoulli Trials” (PDF), “Chapter 9, Section 9.2: Central Limit Theorem for Discrete Independent Trials” (PDF), and “Chapter 9, Section 9.3: Central Limit Theorem for Continuous Independent Trials” (PDF)

Instructions: Read Section 9.1 to Section 9.3 of “Chapter 9: Central Limit Theorem” on pages 325 - 361. This reading further discusses the second fundamental theorem of probability: the central limit theorem.

Reading these textbook sections and taking notes should take approximately 4 hours.

Terms of Use: This resource is licensed under a GNU Free Documentation License (FDL). It is attributed to Charles M. Grinstead, J. Laurie Snell and Dartmouth College.**Assessment: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s**Link: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s*Introduction to Probability*: “Chapter 9, Section 9.1: Exercises”*Introduction to Probability*: “Chapter 9, Section 9.1: Exercises” (PDF)

Instructions: Go to the Section 9.1 exercises on pages 338 - 339 and complete exercises 1, 2, 3, 5, 8, and 14. You can then check your answers to odd-numbered questions here.

Completing this assessment should take approximately 2 hours and 30 minutes.

Terms of Use: This resource is licensed under a GNU Free Documentation License (FDL). It is attributed to Charles M. Grinstead, J. Laurie Snell and Dartmouth College.

**5.3 Other Distributions**
- **Reading: Massachusetts Institute of Technology OpenCourseWare:
Professor Dmitry Panchenko’s Math 18.05: Introduction to
Probability and Statistics: “Lecture 22: Central Limit Theorem,
Gamma Distribution, Beta Distribution”**
Link: Massachusetts Institute of Technology OpenCourseWare:
Professor Dmitry Panchenko’s

*Math 18.05: Introduction to Probability and Statistics*: “Lecture 22: Central Limit Theorem, Gamma Distribution, Beta Distribution” (PDF)

Instructions: Read pages 66 - 69 of this lecture. This reading will give properties of other important distributions, their expected values, etc. - namely, the Gamma and Beta distributions.

Reading this section of the lecture and taking notes should take approximately 2 hours.

Terms of Use: The above material is released under a Creative Commons Attribution-Non Commercial-ShareAlike License 3.0. It is attributed to Dmitry Panchenko and Massachusetts Institute of Technology OpenCourseWare.

**Reading: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s**Link: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s*Math 18.05: Introduction to Probability and Statistics*: “Lecture 26: Confidence Intervals for Parameters of Normal Distribution”*Math 18.05: Introduction to Probability and Statistics*: “Lecture 26: Confidence Intervals for Parameters of Normal Distribution” (PDF)

Instructions: Read pages 78 and 79 of this lecture. This reading will give a simple introduction to other distributions and their properties - namely, the t-distribution and Chi-square distributions, which are widely used in statistics.

Reading this section of the lecture and taking notes should take approximately 2 hours.

Terms of Use: The above material is released under a Creative Commons Attribution-Non Commercial-ShareAlike License 3.0. It is attributed to Dmitry Panchenko and Massachusetts Institute of Technology OpenCourseWare.**Lecture: YouTube: Harvard University: Professor Joseph Blitzstein’s**Link: YouTube: Harvard University: Professor Joseph Blitzstein’s*Statistics 110*: Probability: “Lecture 30: Chi-square, Student-t, Multivariate Normal Distributions”*Statistics 110: Probability*: “Lecture 30: Chi-square, Student-t, Multivariate Normal Distributions” (YouTube)

Instructions: Watch this video for an introduction to other distributions and their properties, namely the Chi-square and t-distributions, which are widely used in statistics. You will see that they are basically offshoots of the normal distribution.

Watching this video and taking notes should take approximately 1 hour and 30 minutes.

Terms of Use: The above material is released under a Creative Commons Attribution-Non Commercial-ShareAlike License 3.0. It is attributed to Harvard University and Dr. Joseph Blitzstein, and the original version can be found here.

**Final Exam**
- **Final Exam: The Saylor Foundation’s “MA252 Final Exam”**
Link: The Saylor Foundation’s “MA252 Final
Exam” (HTML)

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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 on the link.
Completing this assessment should take approximately 1 hour.
```