Unit I: Probability Models And Discrete Random Variables

Course Info

Instructor

Prof. John Tsitsiklis

Departments

Electrical Engineering and Computer Science

As Taught In

Fall 2013

Level

Undergraduate

Topics

Learning Resource Types

Lecture Videos

Recitation Videos

Problem Sets with Solutions

Exams with Solutions

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This unit covers the basic framework of probability theory: probabilistic models, conditional probabilities, independence, the Bayes’ rule, and counting methods. In addition, it introduces discrete random variables and the concept of the Probability Mass Function (PMF) used to describe the probability distribution of one or several random variables. Finally, it defines the concepts of expectation and variance, and their basic properties.

Lecture 1: Probability Models and Axioms

Lecture 2: Conditioning and Bayes’ Rule

Lecture 3: Independence

Lecture 4: Counting

Lecture 5: Discrete Random Variables; Probability Mass Functions; Expectations

Lecture 6: Discrete Random Variable Examples; Joint PMFs

Lecture 7: Multiple Discrete Random Variables

Quiz 1

Looking for something specific in this course? The Resource Index compiles links to most course resources in a single page.

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Lecture Topics

Probability as a mathematical framework for reasoning about uncertainty
Probabilistic models
- Sample space
- Probability law
Axioms of probability
Simple examples

Lecture Activities

Watch the Lecture 1: Probability Models and Axioms Video by Prof. Tsitsiklis (00:51:11)
Review the Lecture 1: Probability Models and Axioms Slides (PDF)
Read Sections 1.1–1.2 in the textbook

Recitation Problems and Recitation Help Videos

Review the recitation problems in the PDF file below and try to solve them on your own. Three of the problems have an accompanying video where a teaching assistant solves the same problem.

Recitation Help Videos

PROBLEM #	PROBLEM TITLE	PROBLEM SOLVED BY
1	The Probability of the Difference of Two Events	Kuang Xu
2	Geniuses and Chocolates	Katie Szeto
4	Uniform Probabilities on a Square	Jimmy Li

Problem Set and Solutions

Work the problems on your own and check your answers when you’re done. Problem set 1 covers Lecture 1.

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Lecture Topics

Conditional probability
Three important tools
- Multiplication rule
- Total probability theorem
- Bayes’ rule

Lecture Activities

Watch the Lecture 2: Conditioning and Bayes’ Rule Video by Prof. Tsitsiklis (00:51:11)
Review the Lecture 2: Conditioning and Bayes’ Rule Slides (PDF)
Read Sections 1.3–1.4 in the textbook

Recitation Problems and Recitation Help Videos

Review the recitation problems in the PDF file below and try to solve them on your own. Three of the problems have an accompanying video where a teaching assistant solves the same problem.

Recitation Help Videos

PROBLEM #	PROBLEM TITLE	PROBLEM SOLVED BY
1	A Coin Tossing Puzzle	Kuang Xu
2	Conditional Probability Example	Katie Szeto
4	The Monty Hall Problem	Jimmy Li

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Lecture Topics

Independence of two events
Independence of a collection of events

Lecture Activities

Watch the Lecture 3: Independence Video by Prof. Tsitsiklis (00:46:30)
Review the Lecture 3: Independence Slides (PDF)
Read Section 1.5 in the textbook

Recitation Problems and Recitation Help Videos

Review the recitation problems in the PDF file below and try to solve them on your own. Two of the problems have an accompanying video where a teaching assistant solves the same problem.

Recitation Help Videos

PROBLEM #	PROBLEM TITLE	PROBLEM SOLVED BY
2	A Random Walker	Kuang Xu
3	Communication over a Noisy Channel	Jimmy Li

Tutorial Problems and Tutorial Help Videos

Review the tutorial problems in the PDF file below and try to solve them on your own. Two of the problems have an accompanying video where a teaching assistant solves the same problem.

Tutorial Help Videos

PROBLEM #	PROBLEM TITLE	PROBLEM SOLVED BY
2	Network Reliability	Kuang Xu
3	A Chess Tournament Problem	Katie Szeto

Problem Set and Solutions

Work the problems on your own and check your answers when you’re done. Problem set 2 covers Lectures 2 and 3.

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Lecture Topics

Principles of Counting
Examples of Counting
- Permutations
- Combinations
- Partitions
Binomial Probabilities

Lecture Activities

Watch the Lecture 4: Counting Video by Prof. Tsitsiklis (00:51:35)
Review the Lecture 4: Counting Slides (PDF)
Read Section 1.6 in the textbook

Recitation Problems and Recitation Help Videos

Review the recitation problems in the PDF file below and try to solve them on your own. Two of the problems have an accompanying video where a teaching assistant solves the same problem.

Recitation Help Videos

PROBLEM #	PROBLEM TITLE	PROBLEM SOLVED BY
2	Rooks on a Chessboard	Katie Szeto
3	Hypergeometric Probabilities	Kuang Xu

Tutorial Problems

Review the tutorial problems in the PDF file below and try to solve them on your own. Check your answers when you’re done.

Tutorial 2 Problems (PDF) Do ONLY problems 1, 3, and 4
Tutorial 2 Solutions (PDF)

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Lecture Topics

Random Variables
Probability of Mass Function (PMF)
Expectation
Variance

Lecture Activities

Watch the Lecture 5: Discrete Random Variables; Probability Mass Functions; Expectations Video by Prof. Tsitsiklis (00:50:35)
Review the Lecture 5: Discrete Random Variables; Probability Mass Functions; Expectations Slides (PDF)
Read Sections 2.1–2.3 and start Section 2.4 in the textbook

Recitation Problems and Recitation Help Videos

Review the recitation problems in the PDF file below and try to solve them on your own. One of the problems has an accompanying video where a teaching assistant solves the same problem.

Recitation Help Videos

PROBLEM #	PROBLEM TITLE	PROBLEM SOLVED BY
3	Sampling People on Buses	Jimmy Li

Tutorial Problems and Tutorial Help Videos

Review the tutorial problems in the PDF file below and try to solve them on your own. One of the problems has an accompanying video where a teaching assistant solves the same problem.

Tutorial 2 Problems (PDF) Do ONLY problem 2
Tutorial 2 Solutions (PDF)

Tutorial Help Videos

PROBLEM #	PROBLEM TITLE	PROBLEM SOLVED BY
2	PMF of a Function of a Random Variable	Katie Szeto

Tutorial 3 Problems (PDF) Do ONLY problem 1
Tutorial 3 Solutions (PDF)

Problem Set and Solutions

Work the problems on your own and check your answers when you’re done. Problem set 3 covers Lectures 4 and 5.

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Lecture Topics

Review: PMF, expectation, variance
Conditional PMF
Geometric PMF
Total expectation theorem
Joint PMF of two random variables

Lecture Activities

Watch the Lecture 6: Discrete Random Variable Examples; Joint PMFs Video by Prof. Tsitsiklis (00:50:53)
Review the Lecture 6: Discrete Random Variable Examples; Joint PMFs Slides (PDF)
Read Sections 2.4–2.6 in the textbook

Recitation Problems and Recitation Help Videos

Review the recitation problems in the PDF file below and try to solve them on your own. Two of the problems have an accompanying video where a teaching assistant solves the same problem.

Recitation Help Videos

PROBLEM #	PROBLEM TITLE	PROBLEM SOLVED BY
1	Flipping a Coin a Random Number of Times	Kuang Xu
2	Joint Probability Mass Function (PMF) Drill 1	Katie Szeto

Tutorial Problems and Tutorial Help Videos

Review the tutorial problems in the PDF file below and try to solve them on your own. One of the problems has an accompanying video where a teaching assistant solves the same problem.

Tutorial 3 Problems (PDF) Do ONLY problem 2
Tutorial 3 Solutions (PDF)

Tutorial Help Videos

PROBLEM #	PROBLEM TITLE	PROBLEM SOLVED BY
2	The Coupon Collector Problem	Kuang Xu

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Lecture Topics

Multiple random variables
- Joint PMF
- Conditioning
- Independence
More on expectations
Binomial distribution revisited
A hat problem

Lecture Activities

Watch the Lecture 7: Multiple Discrete Random Variables Video by Prof. Tsitsiklis (00:50:42)
Review the Lecture 7: Multiple Discrete Random Variables Slides (PDF)
Read Sections 2.6–2.7 in the textbook

Recitation Problems and Recitation Help Videos

Review the recitation problems in the PDF file below and try to solve them on your own. One of the problems has an accompanying video where a teaching assistant solves the same problem.