# MA252: Introduction to Probability Theory

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. One of the most common applications of the binomial distribution is to test items for defects as they come off an assembly line. For the geometric distribution, a common application tests drugs that are known to be effective 70% of the time. We might want to see the probability that the first person on whom the drug is effective this week is the tenth one to take it or the probability that the third person on whom it was effective is the sixth to take it, etc. For the Poisson distribution, typical applications include counting the number of phone calls received by an office in one hour or the number of days school is closed due to snow in a winter.

Completing this unit should take approximately 15 hours.

☐    Subunit 3.1: 1.5 hours

☐    Subunit 3.2: 1.5 hours

☐    Subunit 3.3: 1.5 hours

☐    Subunit 3.4: 10.5 hours

Unit3 Learning Outcomes
Upon successful completion of this unit, you will be able to: - compute probabilities for outcomes of random variables using the binomial and multinomial distributions;
- compute probabilities for outcomes of random variables using the geometric and Poisson distributions; and
- find the mean, variance, and standard deviation for the binomial, multinomial, geometric, and Poisson distributions.

3.1 Binomial Distributions   - Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 10: Binomial Distributions” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences“Lecture 10: Binomial Distributions” (YouTube)

Instructions: Watch this video for an introduction to one of the most common discrete probability distributions: binomial distributions. You will learn how to decide if a distribution is binomial and how to find the mean, variance, and standard deviation of a binomial random variable.

If you perform a sequence of n independent, identical Bernoulli trials - that is, trials with only two possible outcomes: a success and a failure - then the random variable that counts the number of successes is called a binomial random variable. The probability of success is denoted by p, and the probability of failure, which is 1 - p, is denoted by q.

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

3.2 Multinomial Distributions   - Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 12: Multinomial Distributions” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences“Lecture 12: Multinomial Distributions” (YouTube)

Instructions: Watch this video for an introduction to multinomial distributions. This is basically a generalization of the binomial distribution.

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

3.3 Geometric Distributions   - Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 13: Geometric Distributions” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences“Lecture 13: Geometric Distributions” (YouTube)

Instructions: Watch this video for an introduction to multinomial distributions and geometric distributions. The geometric distribution has to do with Bernoulli trials, but it is different from the binomial distribution. This time you perform a sequence of independent, identical Bernoulli trials until you get the first success. The random variable that counts the number of trials needed is called a geometric random variable.

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

3.4 Poisson Distributions   - Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 14: Poisson Distributions” and “Lecture 15: Poisson Distributions (continued)” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences“Lecture 14: Poisson Distributions” (YouTube) and “Lecture 15: Poisson Distributions (continued)” (YouTube)

Instructions: Watch these video lectures for an introduction to Poisson distributions. Poisson experiments are experiments that give the number of outcomes occurring during a given time interval or in a specific region. The random variables that count those outcomes are called Poisson random variables.

For example, a random variable X that counts the number of games postponed due to rain during a football season, or the number of phone calls received per hour by a secretary, are examples of Poisson random variables. Formulas for the distribution will be given as well as the expected value and variance. Specific examples will be given to illustrate the concepts.

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

• Reading: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics: “Lecture 20: Poisson Distribution, Approximation of Binomial Distribution, Normal Distribution” Link: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics“Lecture 20: Poisson Distribution, Approximation of Binomial Distribution, Normal Distribution” (PDF)

Instructions: Read this lecture for a further discussion of Poisson distributions and how to use them to approximate the binomial distribution. The normal distribution, which will be studied in Unit 4, will be briefly introduced, and you will be shown how to use it to approximate the binomial distribution.

Reading this lecture 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 Dmitry Panchenko and Massachusetts Institute of Technology OpenCourseWare.

• Reading: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 5, Section 5.1: Important Distributions” Link: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability“Chapter 5, Section 5.1: Important Distributions” (PDF)

Instructions: Read Section 5.1 “Important Distributions” on pages 183 - 195 in “Chapter 5: Distributions and Densities.” This reading will help you review the distributions of the previous three subunits and learn about the definition of Poisson distributions as well as other distributions, such as the uniform, negative binomial, and hypergeometric 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 3 hours.