# MA252: Introduction to Probability Theory

Unit 4: Continuous Distributions   This unit will introduce you 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.

Completing this unit should take approximately 17 hours.

☐    Subunit 4.1:  3 hours

☐    Subunit 4.2:  3 hours

☐    Subunit 4.3:  2.5 hours

☐    Subunit 4.4:  8.5 hours

Unit4 Learning Outcomes
Upon successful completion of this unit, you will be able to: - define and derive probability density functions of continuous probability distributions; and
- calculate properties of exponential and normal distributions.

4.1 Probability Density Functions   - Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 16: Density Function” and “Lecture 17: Exponential Distributions” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 16: Density Function” (YouTube) and “Lecture 17: Exponential Distributions” (YouTube)

Instructions: Watch these videos for an introduction to probability density functions of continuous random variables. The first video will explain the difference between discrete and continuous random variables and show how density functions are used to study continuous random variables. You will see several nice examples, and you will see again the definition of the expected value, or mean, of a continuous random variable.

The second video is a continuation of the first. It will show again how to get the variance and standard deviation of a continuous random variable. At minute 28, the exponential distribution, which is the subject of the next subunit, will be introduced as an example. It is worthwhile to watch it now. If you want to wait until the next subunit you can stop watching after minute 27.

Watching these videos and taking notes should take approximately 3 hours.

4.2 Exponential Distributions   - Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 17: Exponential Distributions” and “Lecture 18: Normal Distributions” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences“Lecture 17: Exponential Distributions” (YouTube) and “Lecture 18: Normal Distribution” (YouTube)

Instructions: Watch these video lectures for an introduction to common properties of exponential distributions. If you did not watch the last part of Lecture 17 in the previous subunit, please do so now, starting at minute 28. This section of the video defines the exponential distribution and some of its properties, such as the expected value, variance, etc... The second video will show applications of the exponential distributions and present some very nice real-life examples.

Watching these videos and taking notes should take approximately 3 hours.

4.3 Normal Distributions   - Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 19: Normal Distribution (continued)” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences“Lecture 19: Normal Distribution (continued)” (YouTube)

Instructions: Watch this video lecture for an introduction to normal distributions. This is the most important distribution in probability theory, and it is used in almost every field of probability. You will learn about its properties, and you will see its expected value, variance, and standard deviation.

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

• Reading: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics: “Lecture 21: Normal Distribution, Central Limit Theorem” Link: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics“Lecture 21: Normal Distribution, Central Limit Theorem” (PDF)

Instructions: Read the section on normal distribution on pages 63 - 64 of this lecture. This reading will strengthen your understanding of the concepts you saw in the previous video.

Reading this section and taking notes should take approximately 1 hour.

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.

4.4 Standard Normal Distributions   - Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 20: Standard Normal Distributions” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 20: Standard Normal Distributions” (YouTube)

Instructions: Watch this video lecture for an introduction to standard normal distributions and transformations between standard normal distributions and other normal distributions. The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. If you take any normal random variable, subtract its mean from it, and divide the result by its standard deviation, you will get a standard normal random variable. So, we just need to know about standard normal random variables for which probability tables are developed to compute the probabilities that the variable is less than a given value.

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

• Reading: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 5, Section 5.2: Important Densities” Link: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability“Chapter 5, Section 5.2: Important Densities” (PDF)

Instructions: Read Section 5.2 “Important Densities” on pages 205 - 219 in “Chapter 5: Distributions and Densities.” This reading will help you review the distributions of the previous three units and learn about the definition of normal distributions as well as other distributions, such as the uniform, gamma and Chi-square distributions. It is a reinforcement of the concepts you saw in the previous videos and lecture.

Reading this chapter and taking notes should take approximately 4 hours.