Introduction to Probability | Electrical Engineering and Computer Science | MIT OpenCourseWare

Lecture Videos

Brief Introduction

01.1 Lecture Overview

01.10 Interpretations & Uses of Probabilities

01.2 Sample Space

01.3 Sample Space Examples

01.4 Probability Axioms

01.5 Simple Properties of Probabilities

01.6 More Properties of Probabilities

01.7 A Discrete Example

01.8 A Continuous Example

01.9 Countable Additivity

S01.0 Mathematical Background Overview

S01.1 Sets

S01.10 Bonferroni's Inequality

S01.2 De Morgan's Laws

S01.3 Sequences and their Limits

S01.4 When Does a Sequence Converge?

S01.5 Infinite Series

S01.6 The Geometric Series

S01.7 About the Order of Summation in Series with Multiple Indices

S01.8 Countable and Uncountable Sets

S01.9 Proof that a Set of Real Numbers is Uncountable

02.1 Lecture Overview

02.2 Conditional Probabilities

02.3 A Die Roll Example

02.4 Conditional Probabilities Obey the Same Axioms

02.5 A Radar Example and Three Basic Tools

02.6 The Multiplication Rule

02.7 Total Probability Theorem

02.8 Bayes' Rule

03.1 Lecture Overview

03.10 The King's Sibling

03.2 A Coin Tossing Example

03.3 Independence of Two Events

03.4 Independence of Event Complements

03.5 Conditional Independence

03.6 Independence Versus Conditional Independence

03.7 Independence of a Collection of Events

03.8 Independence Versus Pairwise Independence

03.9 Reliability

04.1 Lecture Overview

04.2 The Counting Principle

04.3 Die Roll Example

04.4 Combinations

04.5 Binomial Probabilities

04.6 A Coin Tossing Example

04.7 Partitions

04.8 Each Person Gets An Ace

04.9 Multinomial Probabilities

05.1 Lecture Overview

05.10 The Expected Value Rule

05.11 Linearity of Expectations

05.2 Definition of Random Variables

05.3 Probability Mass Functions

05.4 Bernoulli & Indicator Random Variables

05.5 Uniform Random Variables

05.6 Binomial Random Variables

05.7 Geometric Random Variables

05.8 Expectation

05.9 Elementary Properties of Expectation

S05.1 Supplement: Functions

06.1 Lecture Overview

06.2 Variance

06.3 The Variance of the Bernoulli & the Uniform

06.4 Conditional PMFs & Expectations Given an Event

06.5 Total Expectation Theorem

06.6 Geometric PMF Memorylessness & Expectation

06.7 Joint PMFs and the Expected Value Rule

06.8 Linearity of Expectations & the Mean of the Binomial

07.1 Lecture Overview

07.2 Conditional PMFs

07.2 The Variance of the Geometric

07.3 Conditional Expectation & the Total Expectation Theorem

07.3 Independence of Random Variables Versus Independence of Events

07.4 Independence of Random Variables

07.5 Example

07.6 Independence & Expectations

07.7 Independence, Variances & the Binomial Variance

07.8 The Hat Problem

S07.1 The Inclusion-Exclusion Formula

08.1 Lecture Overview

08.2 Probability Density Functions

08.3 Uniform & Piecewise Constant PDFs

08.4 Means & Variances

08.5 Mean & Variance of the Uniform

08.6 Exponential Random Variables

08.7 Cumulative Distribution Functions

08.8 Normal Random Variables

08.9 Calculation of Normal Probabilities

09.1 Lecture Overview

09.10 Joint CDFs

09.2 Conditioning A Continuous Random Variable on an Event

09.3 Conditioning Example

09.4 Memorylessness of the Exponential PDF

09.5 Total Probability & Expectation Theorems

09.6 Mixed Random Variables

09.7 Joint PDFs

09.8 From The Joint to the Marginal

09.9 Continuous Analogs of Various Properties

S09.1 Buffon's Needle & Monte Carlo Simulation

10.1 Lecture Overview

10.10 Detection of a Binary Signal

10.11 Inference of the Bias of a Coin

10.2 Conditional PDFs

10.3 Comments on Conditional PDFs

10.5 Independence

10.6 Stick-Breaking Example

10.7 Independent Normals

10.8 Bayes Rule Variations

10.9 Mixed Bayes Rule

11.1 Lecture Overview

11.2 The PMF of a Function of a Discrete Random Variable

11.3 A Linear Function of a Continuous Random Variable

11.4 A Linear Function of a Normal Random Variable

11.5 The PDF of a General Function

11.6 The Monotonic Case

11.7 The Intuition for the Monotonic Case

11.8 A Nonmonotonic Example

11.9 The PDF of a Function of Multiple Random Variables

S11.1 Simulation

12.1 Lecture Overview

12.10 Interpreting the Correlation Coefficient

12.11 Correlations Matter

12.2 The Sum of Independent Discrete Random Variables

12.3 The Sum of Independent Continuous Random Variables

12.4 The Sum of Independent Normal Random Variables

12.5 Covariance

12.6 Covariance Properties

12.7 The Variance of the Sum of Random Variables

12.8 The Correlation Coefficient

12.9 Proof of Key Properties of the Correlation Coefficient

13.1 Conditional Expectation Properties

13.1 Lecture Overview

13.10 Mean of the Sum of a Random Number of Random Variables

13.11 Variance of the Sum of a Random Number of Random Variables

13.2 Conditional Expectation as a Random Variable

13.3 The Law of Iterated Expectations

13.4 Stick-Breaking Revisited

13.5 Forecast Revisions

13.6 The Conditional Variance

13.7 Derivation of the Law of Total Variance

13.8 A Simple Example

13.9 Section Means and Variances

14.1 Lecture Overview

14.10 Summary

14.2 Overview of some Application Domains

14.3 Types of Inference Problems

14.4 The Bayesian Inference Framework

14.5 Discrete Parameter, Discrete Observation

14.6 Discrete Parameter, Continuous Observation

14.7 Continuous Parameter, Continuous Observation

14.8 Inferring the Unknown Bias of a Coin and the Beta Distribution

14.9 Inferring the Unknown Bias of a Coin—Point Estimates

S14.1 The Beta Formula

15.1 Lecture Overview

15.2 Recognizing Normal PDFs

15.3 Estimating a Normal Random Variable in the Presence of Additive Noise

15.4 The Case of Multiple Observations

15.5 The Mean Squared Error

15.6 Multiple Parameters; Trajectory Estimation

15.7 Linear Normal Models

15.8 Trajectory Estimation Illustration

16.1 Lecture Overview

16.2 LMS Estimation in the Absence of Observations

16.3 LMS Estimation of One Random Variable Based on Another

16.4 LMS Performance Evaluation

16.5 Example: The LMS Estimate

16.6 Example Continued: LMS Performance Evaluation

16.7 LMS Estimation with Multiple Observations or Unknowns

16.8 Properties of the LMS Estimation Error

17.1 Lecture Overview

17.3 Solution to the LLMS Problem

17.4 Remarks on the LLMS Solution and on the Error Variance

17.5 LLMS Example

17.6 LLMS for Inferring the Parameter of a Coin

17.7 LLMS with Multiple Observations

17.8 The Simplest LLMS Example with Multiple Observations

17.9 The Representation of the Data Matters in LLMS

18.1 Convergence in Probability of the Sum of Two Random Variables

18.1 Lecture Overview

18.10 Total Probability & Total Expectation Theorems

18.2 LLMS Formulation

18.2 The Markov Inequality

18.3 The Chebyshev Inequality

18.4 The Weak Law of Large Numbers

18.5 Polling

18.6 Convergence in Probability

18.7 Convergence in Probability Examples

18.8 Related Topics

S18.2 Jensen's Inequality

S18.3 Hoeffding's Inequality

19.1 Lecture Overview

19.2 The Central Limit Theorem

19.3 Discussion of the CLT

19.4 Illustration of the CLT

19.5 CLT Examples

19.6 Normal Approximation to the Binomial

19.7 Polling Revisited

20.1 Lecture Overview

20.10 Maximum Likelihood Estimation Examples

20.2 Overview of the Classical Statistical Framework

20.3 The Sample Mean and Some Terminology

20.4 On the Mean Squared Error of an Estimator

20.5 Confidence Intervals

20.6 Confidence Intervals for the Estimation of the Mean

20.7 Confidence Intervals for the Mean, When the Variance is Unknown

20.9 Maximum Likelihood Estimation

20.9 Other Natural Estimators

21.1 Lecture Overview

21.10 The Poisson Approximation to the Binomial

21.2 The Bernoulli Process

21.3 Stochastic Processes

21.4 Review of Known Properties of the Bernoulli Process

21.5 The Fresh Start Property

21.6 Example: The Distribution of a Busy Period

21.7 The Time of the K-th Arrival

21.8 Merging of Bernoulli Processes

21.9 Splitting a Bernoulli Process

22.1 Lecture Overview

22.10 An Example

22.2 Definition of the Poisson Process

22.3 Applications of the Poisson Process

22.4 The Poisson PMF for the Number of Arrivals

22.5 The Mean and Variance of the Number of Arrivals

22.6 A Simple Example

22.7 Time of the K–th Arrival

22.8 The Fresh Start Property and Its Implications

22.9 Summary of Results

23.1 Lecture Overview

23.2 The Sum of Independent Poisson Random Variables

23.3 Merging Independent Poisson Processes

23.4 Where is an Arrival of the Merged Process Coming From?

23.5 The Time Until the First (or Last) Lightbulb Burns Out

23.6 Splitting a Poisson Process

23.7 Random Incidence in the Poisson Process

23.8 Random Incidence in a Non–Poisson Process

23.9 Different Sampling Methods Can Give Different Results

S23.1 Poisson Versus Normal Approximations to the Binomial

S23.2 Poisson Arrivals During an Exponential Interval

24.1 Lecture Overview

24.2 Introduction to Markov Processes

24.3 Checkout Counter Example

24.4 Discrete-Time Finite-State Markov Chains

24.5 N–Step Transition Probabilities

24.6 A Numerical Example — Part I

24.6 A Numerical Example — Part III

24.7 Generic Convergence Questions

24.8 Recurrent and Transient States

25.10 Birth-Death Processes — Part I

25.11 Birth-Death Processes — Part II

25.2 Lecture Overview

25.3 Markov Chain Review

25.4 The Probability of a Path

25.5 Recurrent and Transient States: Review

25.6 Periodic States

25.7 Steady-State Probabilities and Convergence

25.8 A Numerical Example — Part II

25.9 Visit Frequency Interpretation of Steady-State Probabilities

26.2 Lecture Overview

26.3 Review of Steady-State Behavior

26.6 Absorption Probabilities

26.6 Design of a Phone System

26.7 Expected Time to Absorption

26.8 Mean First Passage Time

26.9 Gambler's Ruin

Course Info

Instructors

Departments

Electrical Engineering and Computer Science

As Taught In

Spring 2018

Level

Undergraduate

Course Info

Instructors

Departments

Electrical Engineering and Computer Science

As Taught In

Spring 2018

Level

Undergraduate

Topics

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Lecture Videos

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