The videos in Part I introduce the general framework of probability models, multiple discrete or continuous random variables, expectations, conditional distributions, and various powerful tools of general applicability.
The textbook for this subject is Bertsekas, Dimitri, and John Tsitsiklis. Introduction to Probability. 2nd ed. Athena Scientific, 2008. ISBN: 9781886529236.
The authors have made this Selected Summary Material (PDF) available for OCW users.
L = Lecture Content
S = Supplemental Content
SES # & TOPICS | SLIDES |
---|---|
Lecture 1: Probability Models and Axioms | |
› View Lecture Videos L01.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 5MB) L01.2 Sample SpaceFlash and JavaScript are required for this feature. Sample Space > Download from Internet Archive (MP4 - 9MB) L01.3 Sample Space ExamplesFlash and JavaScript are required for this feature. Sample Space Examples > Download from Internet Archive (MP4 - 5MB) L01.4 Probability AxiomsFlash and JavaScript are required for this feature. Probability Axioms > Download from Internet Archive (MP4 - 10MB) L01.5 Simple Properties of ProbabilitiesFlash and JavaScript are required for this feature. Simple Properties of Probabilities > Download from Internet Archive (MP4 - 16MB) L01.6 More Properties of ProbabilitiesFlash and JavaScript are required for this feature. More Properties of Probabilities > Download from Internet Archive (MP4 - 12MB) L01.7 A Discrete ExampleFlash and JavaScript are required for this feature. A Discrete Example > Download from Internet Archive (MP4 - 7MB) L01.8 A Continuous ExampleFlash and JavaScript are required for this feature. A Continuous Example > Download from Internet Archive (MP4 - 5MB) L01.9 Countable AdditivityFlash and JavaScript are required for this feature. Countable Additivity > Download from Internet Archive (MP4 - 13MB) L01.10 Interpretations & Uses of ProbabilitiesFlash and JavaScript are required for this feature. Interpretations & Uses of Probabilities > Download from Internet Archive (MP4 - 4MB) |
|
Lecture 1 Supplement: Mathematical Background | |
› View Lecture Videos S01.0 Mathematical Background OverviewFlash and JavaScript are required for this feature. Mathematical Background Overview > Download from Internet Archive (MP4 - 2MB) S01.1 SetsFlash and JavaScript are required for this feature. Sets > Download from Internet Archive (MP4 - 14MB) S01.2 De Morgan's LawsFlash and JavaScript are required for this feature. De Morgan's Laws > Download from Internet Archive (MP4 - 7MB) S01.3 Sequences and their LimitsFlash and JavaScript are required for this feature. Sequences and their Limits > Download from Internet Archive (MP4 - 8MB) S01.4 When Does a Sequence Converge?Flash and JavaScript are required for this feature. When Does a Sequence Converge? > Download from Internet Archive (MP4 - 3MB) S01.5 Infinite SeriesFlash and JavaScript are required for this feature. Infinite Series > Download from Internet Archive (MP4 - 4MB) S01.6 The Geometric SeriesFlash and JavaScript are required for this feature. The Geometric Series > Download from Internet Archive (MP4 - 5MB) S01.7 About the Order of Summation in Series with Multiple IndicesFlash and JavaScript are required for this feature. About the Order of Summation in Series with Multiple Indices > Download from Internet Archive (MP4 - 12MB) S01.8 Countable and Uncountable SetsFlash and JavaScript are required for this feature. Countable and Uncountable Sets > Download from Internet Archive (MP4 - 8MB) S01.9 Proof that a Set of Real Numbers is UncountableFlash and JavaScript are required for this feature. Proof that a Set of Real Numbers is Uncountable > Download from Internet Archive (MP4 - 5MB) S01.10 Bonferroni's InequalityFlash and JavaScript are required for this feature. Bonferroni's Inequality > Download from Internet Archive (MP4 - 12MB) |
|
Lecture 2: Conditioning and Bayes' Rule | |
› View Lecture Videos L02.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 5MB) L02.2 Conditional ProbabilitiesFlash and JavaScript are required for this feature. Conditional Probabilities > Download from Internet Archive (MP4 - 10MB) L02.3 A Die Roll ExampleFlash and JavaScript are required for this feature. A Die Roll Example > Download from Internet Archive (MP4 - 6MB) L02.4 Conditional Probabilities Obey the Same AxiomsFlash and JavaScript are required for this feature. Conditional Probabilities Obey the Same Axioms > Download from Internet Archive (MP4 - 10MB) L02.5 A Radar Example and Three Basic ToolsFlash and JavaScript are required for this feature. A Radar Example and Three Basic Tools > Download from Internet Archive (MP4 - 13) L02.6 The Multiplication RuleFlash and JavaScript are required for this feature. The Multiplication Rule > Download from Internet Archive (MP4 - 8MB) L02.7 Total Probability TheoremFlash and JavaScript are required for this feature. Total Probability Theorem > Download from Internet Archive (MP4 - 9MB) L02.8 Bayes' RuleFlash and JavaScript are required for this feature. Bayes' Rule > Download from Internet Archive (MP4 - 5MB) |
|
Lecture 3: Independence | |
› View Lecture Videos L03.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 4MB) L03.2 A Coin Tossing ExampleFlash and JavaScript are required for this feature. A Coin Tossing Example > Download from Internet Archive (MP4 - 14MB) L03.3 Independence of Two EventsFlash and JavaScript are required for this feature. Independence of Two Events > Download from Internet Archive (MP4 - 7MB) L03.4 Independence of Event ComplementsFlash and JavaScript are required for this feature. Independence of Event Complements > Download from Internet Archive (MP4 - 4MB) L03.5 Conditional IndependenceFlash and JavaScript are required for this feature. Conditional Independence > Download from Internet Archive (MP4 - 4MB) L03.6 Independence Versus Conditional IndependenceFlash and JavaScript are required for this feature. Independence Versus Conditional Independence > Download from Internet Archive (MP4 - 8MB) L03.7 Independence of a Collection of EventsFlash and JavaScript are required for this feature. Independence of a Collection of Events > Download from Internet Archive (MP4 - 9MB) L03.8 Independence Versus Pairwise IndependenceFlash and JavaScript are required for this feature. Independence Versus Pairwise Independence > Download from Internet Archive (MP4 - 13MB) L03.9 ReliabilityFlash and JavaScript are required for this feature. Reliability > Download from Internet Archive (MP4 - 9MB) L03.10 The King's SiblingFlash and JavaScript are required for this feature. The King's Sibling > Download from Internet Archive (MP4 - 8MB) |
|
Lecture 4: Counting | |
› View Lecture Videos L04.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 6MB) L04.2 The Counting PrincipleFlash and JavaScript are required for this feature. The Counting Principle > Download from Internet Archive (MP4 - 13MB) L04.3 Die Roll ExampleFlash and JavaScript are required for this feature. Die Roll Example > Download from Internet Archive (MP4 - 6MB) L04.4 CombinationsFlash and JavaScript are required for this feature. Combinations > Download from Internet Archive (MP4 - 12MB) L04.5 Binomial ProbabilitiesFlash and JavaScript are required for this feature. Binomial Probabilities > Download from Internet Archive (MP4 - 8MB) L04.6 A Coin Tossing ExampleFlash and JavaScript are required for this feature. A Coin Tossing Example > Download from Internet Archive (MP4 - 15MB) L04.7 PartitionsFlash and JavaScript are required for this feature. Partitions > Download from Internet Archive (MP4 - 7MB) L04.8 Each Person Gets An AceFlash and JavaScript are required for this feature. Each Person Gets An Ace > Download from Internet Archive (MP4 - 11MB) L04.9 Multinomial ProbabilitiesFlash and JavaScript are required for this feature. Multinomial Probabilities > Download from Internet Archive (MP4 - 13MB) |
|
Lecture 5: Discrete Random Variables Part I | |
› View Lecture Videos L05.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 4MB) L05.2 Definition of Random VariablesFlash and JavaScript are required for this feature. Definition of Random Variables > Download from Internet Archive (MP4 - 12MB) L05.3 Probability Mass FunctionsFlash and JavaScript are required for this feature. Probability Mass Functions > Download from Internet Archive (MP4 - 15MB) L05.4 Bernoulli & Indicator Random VariablesFlash and JavaScript are required for this feature. Bernoulli & Indicator Random Variables > Download from Internet Archive (MP4 - 5MB) L05.5 Uniform Random VariablesFlash and JavaScript are required for this feature. Uniform Random Variables > Download from Internet Archive (MP4 - 5MB) L05.6 Binomial Random VariablesFlash and JavaScript are required for this feature. Binomial Random Variables > Download from Internet Archive (MP4 - 11MB) L05.7 Geometric Random VariablesFlash and JavaScript are required for this feature. Geometric Random Variables > Download from Internet Archive (MP4 - 10MB) L05.8 ExpectationFlash and JavaScript are required for this feature. Expectation > Download from Internet Archive (MP4 - 14MB) L05.9 Elementary Properties of ExpectationFlash and JavaScript are required for this feature. Elementary Properties of Expectation > Download from Internet Archive (MP4 - 5MB) L05.10 The Expected Value RuleFlash and JavaScript are required for this feature. The Expected Value Rule > Download from Internet Archive (MP4 - 14MB) L05.11 Linearity of ExpectationsFlash and JavaScript are required for this feature. Linearity of Expectations > Download from Internet Archive (MP4 - 6MB) S05.1 Supplement: FunctionsFlash and JavaScript are required for this feature. Supplement: Functions > Download from Internet Archive (MP4 - 11MB) |
|
Lecture 6: Discrete Random Variables Part II | |
› View Lecture Videos L06.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 5MB) L06.2 VarianceFlash and JavaScript are required for this feature. Variance > Download from Internet Archive (MP4 - 16MB) L06.3 The Variance of the Bernoulli & the UniformFlash and JavaScript are required for this feature. The Variance of the Bernoulli & the Uniform > Download from Internet Archive (MP4 - 10MB) L06.4 Conditional PMFs & Expectations Given an EventFlash and JavaScript are required for this feature. Conditional PMFs & Expectations Given an Event > Download from Internet Archive (MP4 - 9MB) L06.5 Total Expectation TheoremFlash and JavaScript are required for this feature. Total Expectation Theorem > Download from Internet Archive (MP4 - 8MB) L06.6 Geometric PMF Memorylessness & ExpectationFlash and JavaScript are required for this feature. Geometric PMF Memorylessness & Expectation > Download from Internet Archive (MP4 - 13MB) L06.7 Joint PMFs and the Expected Value RuleFlash and JavaScript are required for this feature. Joint PMFs and the Expected Value Rule > Download from Internet Archive (MP4 - 13MB) L06.8 Linearity of Expectations & the Mean of the BinomialFlash and JavaScript are required for this feature. Linearity of Expectations & the Mean of the Binomial > Download from Internet Archive (MP4 - 11MB) |
|
Lecture 7: Discrete Random Variables Part III | |
› View Lecture Videos L07.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 4MB) L07.2 Conditional PMFsFlash and JavaScript are required for this feature. Conditional PMFs > Download from Internet Archive (MP4 - 14MB) L07.3 Conditional Expectation & the Total Expectation TheoremFlash and JavaScript are required for this feature. Conditional Expectation & the Total Expectation Theorem > Download from Internet Archive (MP4 - 7MB) L07.4 Independence of Random VariablesFlash and JavaScript are required for this feature. Independence of Random Variables > Download from Internet Archive (MP4 - 7MB) L07.5 ExampleFlash and JavaScript are required for this feature. Example > Download from Internet Archive (MP4 - 6MB) L07.6 Independence & ExpectationsFlash and JavaScript are required for this feature. Independence & Expectations > Download from Internet Archive (MP4 - 6MB) L07.7 Independence, Variances & the Binomial VarianceFlash and JavaScript are required for this feature. Independence, Variances & the Binomial Variance > Download from Internet Archive (MP4 - 9MB) L07.8 The Hat ProblemFlash and JavaScript are required for this feature. The Hat Problem > Download from Internet Archive (MP4 - 21MB) S07.1 The Inclusion-Exclusion FormulaFlash and JavaScript are required for this feature. The Inclusion-Exclusion Formula > Download from Internet Archive (MP4 - 18MB) S07.2 The Variance of the GeometricFlash and JavaScript are required for this feature. The Variance of the Geometric > Download from Internet Archive (MP4 - 8MB) S07.3 Independence of Random Variables Versus Independence of EventsFlash and JavaScript are required for this feature. Independence of Random Variables Versus Independence of Events > Download from Internet Archive (MP4 - 10MB) |
|
Lecture 8: Continuous Random Variables Part I | |
› View Lecture Videos L08.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 3MB) L08.2 Probability Density FunctionsFlash and JavaScript are required for this feature. Probability Density Functions > Download from Internet Archive (MP4 - 13MB) L08.3 Uniform & Piecewise Constant PDFsFlash and JavaScript are required for this feature. Uniform & Piecewise Constant PDFs > Download from Internet Archive (MP4 - 3MB) L08.4 Means & VariancesFlash and JavaScript are required for this feature. Means & Variances > Download from Internet Archive (MP4 - 9MB) L08.5 Mean & Variance of the UniformFlash and JavaScript are required for this feature. Mean & Variance of the Uniform > Download from Internet Archive (MP4 - 5MB) L08.6 Exponential Random VariablesFlash and JavaScript are required for this feature. Exponential Random Variables > Download from Internet Archive (MP4 - 11MB) L08.7 Cumulative Distribution FunctionsFlash and JavaScript are required for this feature. Cumulative Distribution Functions > Download from Internet Archive (MP4 - 15MB) L08.8 Normal Random VariablesFlash and JavaScript are required for this feature. Normal Random Variables > Download from Internet Archive (MP4 - 10MB) L08.9 Calculation of Normal ProbabilitiesFlash and JavaScript are required for this feature. Calculation of Normal Probabilities > Download from Internet Archive (MP4 - 19MB) |
|
Lecture 9: Continuous Random Variables Part II | |
› View Lecture Videos L09.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 4MB) L09.2 Conditioning A Continuous Random Variable on an EventFlash and JavaScript are required for this feature. Conditioning A Continuous Random Variable on an Event > Download from Internet Archive (MP4 - 12MB) L09.3 Conditioning ExampleFlash and JavaScript are required for this feature. Conditioning Example > Download from Internet Archive (MP4 - 4MB) L09.4 Memorylessness of the Exponential PDFFlash and JavaScript are required for this feature. Memorylessness of the Exponential PDF > Download from Internet Archive (MP4 - 9MB) L09.5 Total Probability & Expectation TheoremsFlash and JavaScript are required for this feature. Total Probability & Expectation Theorems > Download from Internet Archive (MP4 - 9MB) L09.6 Mixed Random VariablesFlash and JavaScript are required for this feature. Mixed Random Variables > Download from Internet Archive (MP4 - 7MB) L09.7 Joint PDFsFlash and JavaScript are required for this feature. Joint PDFs > Download from Internet Archive (MP4 - 11MB) L09.8 From The Joint to the MarginalFlash and JavaScript are required for this feature. From The Joint to the Marginal > Download from Internet Archive (MP4 - 9MB) L09.9 Continuous Analogs of Various PropertiesFlash and JavaScript are required for this feature. Continuous Analogs of Various Properties > Download from Internet Archive (MP4 - 2MB) L09.10 Joint CDFsFlash and JavaScript are required for this feature. Joint CDFs > Download from Internet Archive (MP4 - 6MB) S09.1 Buffon's Needle & Monte Carlo SimulationFlash and JavaScript are required for this feature. Buffon's Needle & Monte Carlo Simulation > Download from Internet Archive (MP4 - 19MB) |
|
Lecture 10: Continuous Random Variables Part III | |
› View Lecture Videos L10.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 4MB) L10.2 Conditional PDFsFlash and JavaScript are required for this feature. Conditional PDFs > Download from Internet Archive (MP4 - 8MB) L10.3 Comments on Conditional PDFsFlash and JavaScript are required for this feature. Comments on Conditional PDFs > Download from Internet Archive (MP4 - 7MB) L10.4 Total Probability & Total Expectation TheoremsFlash and JavaScript are required for this feature. Total Probability & Total Expectation Theorems > Download from Internet Archive (MP4 - 7MB) L10.5 IndependenceFlash and JavaScript are required for this feature. Independence > Download from Internet Archive (MP4 - 4MB) L10.6 Stick-Breaking ExampleFlash and JavaScript are required for this feature. Stick-Breaking Example > Download from Internet Archive (MP4 - 16MB) L10.7 Independent NormalsFlash and JavaScript are required for this feature. Independent Normals > Download from Internet Archive (MP4 - 7MB) L10.8 Bayes Rule VariationsFlash and JavaScript are required for this feature. Bayes Rule Variations > Download from Internet Archive (MP4 - 4MB) L10.9 Mixed Bayes RuleFlash and JavaScript are required for this feature. Mixed Bayes Rule > Download from Internet Archive (MP4 - 11MB) L10.10 Detection of a Binary SignalFlash and JavaScript are required for this feature. Detection of a Binary Signal > Download from Internet Archive (MP4 - 13MB) L10.11 Inference of the Bias of a CoinFlash and JavaScript are required for this feature. Inference of the Bias of a Coin > Download from Internet Archive (MP4 - 8MB) |
|
Lecture 11: Derived Distributions | |
› View Lecture Videos L11.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 5MB) L11.2 The PMF of a Function of a Discrete Random VariableFlash and JavaScript are required for this feature. The PMF of a Function of a Discrete Random Variable > Download from Internet Archive (MP4 - 7MB) L11.3 A Linear Function of a Continuous Random VariableFlash and JavaScript are required for this feature. A Linear Function of a Continuous Random Variable > Download from Internet Archive (MP4 - 14MB) L11.4 A Linear Function of a Normal Random VariableFlash and JavaScript are required for this feature. A Linear Function of a Normal Random Variable > Download from Internet Archive (MP4 - 4MB) L11.5 The PDF of a General FunctionFlash and JavaScript are required for this feature. The PDF of a General Function > Download from Internet Archive (MP4 - 12MB) L11.6 The Monotonic CaseFlash and JavaScript are required for this feature. The Monotonic Case > Download from Internet Archive (MP4 - 14MB) L11.7 The Intuition for the Monotonic CaseFlash and JavaScript are required for this feature. The Intuition for the Monotonic Case > Download from Internet Archive (MP4 - 6MB) L11.8 A Nonmonotonic ExampleFlash and JavaScript are required for this feature. A Nonmonotonic Example > Download from Internet Archive (MP4 - 10MB) L11.9 The PDF of a Function of Multiple Random VariablesFlash and JavaScript are required for this feature. The PDF of a Function of Multiple Random Variables > Download from Internet Archive (MP4 - 10MB) S11.1 SimulationFlash and JavaScript are required for this feature. Simulation > Download from Internet Archive (MP4 - 17MB) |
|
Lecture 12: Sum of Independent R.V.s. Covariance and Correlation | |
› View Lecture Videos L12.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 4MB) L12.2 The Sum of Independent Discrete Random VariablesFlash and JavaScript are required for this feature. The Sum of Independent Discrete Random Variables > Download from Internet Archive (MP4 - 10MB) L12.3 The Sum of Independent Continuous Random VariablesFlash and JavaScript are required for this feature. The Sum of Independent Continuous Random Variables > Download from Internet Archive (MP4 - 9MB) L12.4 The Sum of Independent Normal Random VariablesFlash and JavaScript are required for this feature. The Sum of Independent Normal Random Variables > Download from Internet Archive (MP4 - 4MB) L12.5 CovarianceFlash and JavaScript are required for this feature. Covariance > Download from Internet Archive (MP4 - 7MB) L12.6 Covariance PropertiesFlash and JavaScript are required for this feature. Covariance Properties > Download from Internet Archive (MP4 - 8MB) L12.7 The Variance of the Sum of Random VariablesFlash and JavaScript are required for this feature. The Variance of the Sum of Random Variables > Download from Internet Archive (MP4 - 9MB) L12.8 The Correlation CoefficientFlash and JavaScript are required for this feature. The Correlation Coefficient > Download from Internet Archive (MP4 - 8MB) L12.9 Proof of Key Properties of the Correlation CoefficientFlash and JavaScript are required for this feature. Proof of Key Properties of the Correlation Coefficient > Download from Internet Archive (MP4 - 5MB) L12.10 Interpreting the Correlation CoefficientFlash and JavaScript are required for this feature. Interpreting the Correlation Coefficient > Download from Internet Archive (MP4 - 12MB) L12.11 Correlations MatterFlash and JavaScript are required for this feature. Correlations Matter > Download from Internet Archive (MP4 - 8MB) |
|
Lecture 13: Conditional Expectation & Variance Revisited; Sum of a Random Number of Independent R.V.s | |
› View Lecture Videos L13.1 Lecture OverviewFlash and JavaScript are required for this feature. Lecture Overview > Download from Internet Archive (MP4 - 4MB) L13.2 Conditional Expectation as a Random VariableFlash and JavaScript are required for this feature. Conditional Expectation as a Random Variable > Download from Internet Archive (MP4 - 5MB) L13.3 The Law of Iterated ExpectationsFlash and JavaScript are required for this feature. The Law of Iterated Expectations > Download from Internet Archive (MP4 - 5MB) L13.4 Stick-Breaking RevisitedFlash and JavaScript are required for this feature. Stick-Breaking Revisited > Download from Internet Archive (MP4 - 5MB) L13.5 Forecast RevisionsFlash and JavaScript are required for this feature. Forecast Revisions > Download from Internet Archive (MP4 - 6MB) L13.6 The Conditional VarianceFlash and JavaScript are required for this feature. The Conditional Variance > Download from Internet Archive (MP4 - 5MB) L13.7 Derivation of the Law of Total VarianceFlash and JavaScript are required for this feature. Derivation of the Law of Total Variance > Download from Internet Archive (MP4 - 7MB) L13.8 A Simple ExampleFlash and JavaScript are required for this feature. A Simple Example > Download from Internet Archive (MP4 - 8MB) L13.9 Section Means and VariancesFlash and JavaScript are required for this feature. Section Means and Variances > Download from Internet Archive (MP4 - 11MB) L13.10 Mean of the Sum of a Random Number of Random VariablesFlash and JavaScript are required for this feature. Mean of the Sum of a Random Number of Random Variables > Download from Internet Archive (MP4 - 8MB) L13.11 Variance of the Sum of a Random Number of Random VariablesFlash and JavaScript are required for this feature. Variance of the Sum of a Random Number of Random Variables > Download from Internet Archive (MP4 - 7MB) S13.1 Conditional Expectation PropertiesFlash and JavaScript are required for this feature. Conditional Expectation Properties > Download from Internet Archive (MP4 - 10MB) |