MIT RES.6-012 Introduction to Probability, Spring 2018
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1 modules • 266 lessons • 29.5 hours of video
MIT RES.6-012 Introduction to Probability, Spring 2018
266 lessons
• 29.5 hours
MIT RES.6-012 Introduction to Probability, Spring 2018
266 lessons
• 29.5 hours
- L01.1 Lecture Overview 01:52
- L01.2 Sample Space 05:38
- L01.3 Sample Space Examples 05:03
- L01.4 Probability Axioms 08:55
- L01.5 Simple Properties of Probabilities 11:05
- L01.6 More Properties of Probabilities 08:40
- L01.7 A Discrete Example 05:13
- L01.8 A Continuous Example 05:20
- L01.9 Countable Additivity 12:10
- L01.10 Interpretations & Uses of Probabilities 03:48
- S01.0 Mathematical Background Overview 01:25
- S01.1 Sets 10:55
- S01.2 De Morgan's Laws 04:53
- S01.3 Sequences and their Limits 06:00
- S01.4 When Does a Sequence Converge 02:46
- S01.5 Infinite Series 03:11
- S01.6 The Geometric Series 04:07
- S01.7 About the Order of Summation in Series with Multiple Indices 10:05
- S01.8 Countable and Uncountable Sets 06:19
- S01.9 Proof That a Set of Real Numbers is Uncountable 04:02
- S01.10 Bonferroni's Inequality 09:28
- L02.1 Lecture Overview 02:07
- L02.2 Conditional Probabilities 09:00
- L02.3 A Die Roll Example 05:02
- L02.4 Conditional Probabilities Obey the Same Axioms 07:45
- L02.5 A Radar Example and Three Basic Tools 10:59
- L02.6 The Multiplication Rule 06:17
- L02.7 Total Probability Theorem 05:25
- L02.8 Bayes' Rule 04:28
- L03.1 Lecture Overview 01:27
- L03.2 A Coin Tossing Example 07:59
- L03.3 Independence of Two Events 06:10
- L03.4 Independence of Event Complements 02:59
- L03.5 Conditional Independence 02:46
- L03.6 Independence Versus Conditional Independence 05:30
- L03.7 Independence of a Collection of Events 06:00
- L03.8 Independence Versus Pairwise Independence 08:35
- L03.9 Reliability 07:28
- L03.10 The King's Sibling 06:54
- L04.1 Lecture Overview 02:29
- L04.2 The Counting Principle 11:12
- L04.3 Die Roll Example 04:39
- L04.4 Combinations 10:08
- L04.5 Binomial Probabilities 06:38
- L04.6 A Coin Tossing Example 11:48
- L04.7 Partitions 05:20
- L04.8 Each Person Gets An Ace 09:45
- L04.9 Multinomial Probabilities 10:36
- L05.1 Lecture Overview 01:40
- L05.2 Definition of Random Variables 09:14
- L05.3 Probability Mass Functions 10:21
- L05.4 Bernoulli & Indicator Random Variables 03:06
- L05.5 Uniform Random Variables 04:06
- L05.6 Binomial Random Variables 06:08
- L05.7 Geometric Random Variables 07:37
- L05.8 Expectation 10:38
- L05.9 Elementary Properties of Expectation 04:12
- L05.10 The Expected Value Rule 10:00
- L05.11 Linearity of Expectations 03:59
- S05.1 Supplement: Functions 08:08
- L06.1 Lecture Overview 02:02
- L06.2 Variance 10:43
- L06.3 The Variance of the Bernoulli & The Uniform 08:40
- L06.4 Conditional PMFs & Expectations Given an Event 07:31
- L06.5 Total Expectation Theorem 06:28
- L06.6 Geometric PMF Memorylessness & Expectation 10:29
- L06.7 Joint PMFs and the Expected Value Rule 10:16
- L06.8 Linearity of Expectations & The Mean of the Binomial 08:25
- L07.1 Lecture Overview 01:50
- L07.2 Conditional PMFs 10:48
- L07.3 Conditional Expectation & the Total Expectation Theorem 06:10
- L07.4 Independence of Random Variables 05:08
- L07.5 Example 04:44
- L07.6 Independence & Expectations 04:22
- L07.7 Independence, Variances & the Binomial Variance 07:09
- L07.8 The Hat Problem 16:09
- S07.1 The Inclusion-Exclusion Formula 11:13
- S07.2 The Variance of the Geometric 05:42
- S07.3 Independence of Random Variables Versus Independence of Events 06:51
- L08.1 Lecture Overview 01:13
- L08.2 Probability Density Functions 11:09
- L08.3 Uniform & Piecewise Constant PDFs 02:52
- L08.4 Means & Variances 06:57
- L08.5 Mean & Variance of the Uniform 03:56
- L08.6 Exponential Random Variables 08:09
- L08.7 Cumulative Distribution Functions 12:48
- L08.8 Normal Random Variables 09:14
- L08.9 Calculation of Normal Probabilities 10:11
- L09.1 Lecture Overview 01:33
- L09.2 Conditioning A Continuous Random Variable on an Event 09:56
- L09.3 Conditioning Example 03:08
- L09.4 Memorylessness of the Exponential PDF 08:18
- L09.5 Total Probability & Expectation Theorems 06:51
- L09.6 Mixed Random Variables 05:35
- L09.7 Joint PDFs 09:18
- L09.8 From The Joint to the Marginal 07:23
- L09.9 Continuous Analogs of Various Properties 01:40
- L09.10 Joint CDFs 04:16
- S09.1 Buffon's Needle & Monte Carlo Simulation 16:12
- L10.1 Lecture Overview 01:42
- L10.2 Conditional PDFs 06:57
- L10.3 Comments on Conditional PDFs 04:34
- L10.4 Total Probability & Total Expectation Theorems 05:17
- L10.5 Independence 03:35
- L10.6 Stick-Breaking Example 10:02
- L10.7 Independent Normals 05:36
- L10.8 Bayes Rule Variations 03:27
- L10.9 Mixed Bayes Rule 08:33
- L10.10 Detection of a Binary Signal 09:15
- L10.11 Inference of the Bias of a Coin 06:00
- L11.1 Lecture Overview 01:52
- L11.2 The PMF of a Function of a Discrete Random Variable 06:42
- L11.3 A Linear Function of a Continuous Random Variable 11:18
- L11.4 A Linear Function of a Normal Random Variable 02:45
- L11.5 The PDF of a General Function 09:47
- L11.6 The Monotonic Case 11:07
- L11.7 The Intuition for the Monotonic Case 05:28
- L11.8 A Nonmonotonic Example 07:14
- L11.9 The PDF of a Function of Multiple Random Variables 07:42
- S11.1 Simulation 12:35
- L12.1 Lecture Overview 01:29
- L12.2 The Sum of Independent Discrete Random Variables 07:52
- L12.3 The Sum of Independent Continuous Random Variables 06:45
- L12.4 The Sum of Independent Normal Random Variables 03:10
- L12.5 Covariance 05:54
- L12.6 Covariance Properties 05:48
- L12.7 The Variance of the Sum of Random Variables 05:36
- L12.8 The Correlation Coefficient 07:03
- L12.9 Proof of Key Properties of the Correlation Coefficient 03:52
- L12.10 Interpreting the Correlation Coefficient 05:50
- L12.11 Correlations Matter 06:22
- L13.1 Lecture Overview 01:47
- L13.2 Conditional Expectation as a Random Variable 04:31
- L13.3 The Law of Iterated Expectations 03:58
- L13.4 Stick-Breaking Revisited 03:53
- L13.5 Forecast Revisions 04:38
- L13.6 The Conditional Variance 05:02
- L13.7 Derivation of the Law of Total Variance 04:54
- L13.8 A Simple Example 06:29
- L13.9 Section Means and Variances 09:04
- L13.10 Mean of the Sum of a Random Number of Random Variables 06:26
- L13.11 Variance of the Sum of a Random Number of Random Variables 05:10
- S13.1 Conditional Expectation Properties 08:13
- L14.1 Lecture Overview 02:10
- L14.2 Overview of Some Application Domains 05:17
- L14.3 Types of Inference Problems 05:24
- L14.4 The Bayesian Inference Framework 09:48
- L14.5 Discrete Parameter, Discrete Observation 06:46
- L14.6 Discrete Parameter, Continuous Observation 04:35
- L14.7 Continuous Parameter, Continuous Observation 03:46
- L14.8 Inferring the Unknown Bias of a Coin and the Beta Distribution 07:35
- L14.9 Inferring the Unknown Bias of a Coin - Point Estimates 09:30
- L14.10 Summary 05:41
- S14.1 The Beta Formula 10:24
- L15.1 Lecture Overview 01:59
- L15.2 Recognizing Normal PDFs 07:15
- L15.3 Estimating a Normal Random Variable in the Presence of Additive Noise 08:18
- L15.4 The Case of Multiple Observations 13:47
- L15.5 The Mean Squared Error 13:02
- L15.6 Multiple Parameters; Trajectory Estimation 10:32
- L15.7 Linear Normal Models 05:12
- L15.8 Trajectory Estimation Illustration 10:55
- L16.1 Lecture Overview 01:13
- L16.2 LMS Estimation in the Absence of Observations 06:48
- L16.3 LMS Estimation of One Random Variable Based on Another 09:24
- L16.4 LMS Performance Evaluation 04:32
- L16.5 Example: The LMS Estimate 06:31
- L16.6 Example Continued: LMS Performance Evaluation 05:29
- L16.7 LMS Estimation with Multiple Observations or Unknowns 05:24
- L16.8 Properties of the LMS Estimation Error 05:59
- L17.1 Lecture Overview 01:41
- L17.2 LLMS Formulation 04:58
- L17.3 Solution to the LLMS Problem 05:06
- L17.4 Remarks on the LLMS Solution and on the Error Variance 08:02
- L17.5 LLMS Example 06:43
- L17.6 LLMS for Inferring the Parameter of a Coin 11:29
- L17.7 LLMS with Multiple Observations 06:54
- L17.8 The Simplest LLMS Example with Multiple Observations 05:06
- L17.9 The Representation of the Data Matters in LLMS 07:03
- L18.1 Lecture Overview 01:57
- L18.2 The Markov Inequality 10:21
- L18.3 The Chebyshev Inequality 05:57
- L18.4 The Weak Law of Large Numbers 07:31
- L18.5 Polling 08:12
- L18.6 Convergence in Probability 08:28
- L18.7 Convergence in Probability Examples 08:05
- L18.8 Related Topics 06:44
- S18.1 Convergence in Probability of the Sum of Two Random Variables 10:13
- S18.2 Jensen's Inequality 12:19
- S18.3 Hoeffding's Inequality 18:28
- L19.1 Lecture Overview 01:50
- L19.2 The Central Limit Theorem 06:58
- L19.3 Discussion of the CLT 09:00
- L19.4 Illustration of the CLT 02:54
- L19.5 CLT Examples 13:56
- L19.6 Normal Approximation to the Binomial 11:53
- L19.7 Polling Revisited 13:54
- L20.1 Lecture Overview 02:46
- L20.2 Overview of the Classical Statistical Framework 11:00
- L20.3 The Sample Mean and Some Terminology 04:58
- L20.4 On the Mean Squared Error of an Estimator 06:53
- L20.5 Confidence Intervals 05:04
- L20.6 Confidence Intervals for the Estimation of the Mean 04:27
- L20.7 Confidence Intervals for the Mean, When the Variance is Unknown 06:13
- L20.8 Other Natural Estimators 04:37
- L20.9 Maximum Likelihood Estimation 06:32
- L20.10 Maximum Likelihood Estimation Examples 10:20
- L21.1 Lecture Overview 02:01
- L21.2 The Bernoulli Process 04:21
- L21.3 Stochastic Processes 06:21
- L21.4 Review of Known Properties of the Bernoulli Process 02:20
- L21.5 The Fresh Start Property 11:26
- L21.6 Example: The Distribution of a Busy Period 04:16
- L21.7 The Time of the K-th Arrival 08:12
- L21.8 Merging of Bernoulli Processes 07:12
- L21.9 Splitting a Bernoulli Process 05:54
- L21.10 The Poisson Approximation to the Binomial 06:12
- L22.1 Lecture Overview 01:31
- L22.2 Definition of the Poisson Process 05:07
- L22.3 Applications of the Poisson Process 03:03
- L22.4 The Poisson PMF for the Number of Arrivals 08:01
- L22.5 The Mean and Variance of the Number of Arrivals 03:22
- L22.6 A Simple Example 03:07
- L22.7 Time of the K-th Arrival 10:41
- L22.8 The Fresh Start Property and Its Implications 10:33
- L22.9 Summary of Results 02:34
- L22.10 An Example 14:08
- L23.1 Lecture Overview 01:39
- L23.2 The Sum of Independent Poisson Random Variables 04:03
- L23.3 Merging Independent Poisson Processes 08:22
- L23.4 Where is an Arrival of the Merged Process Coming From? 05:00
- L23.5 The Time Until the First (or last) Lightbulb Burns Out 11:25
- L23.6 Splitting a Poisson Process 05:06
- L23.7 Random Incidence in the Poisson Process 09:09
- L23.8 Random Incidence in a Non-Poisson Process 04:36
- L23.9 Different Sampling Methods can Give Different Results 03:59
- S23.1 Poisson Versus Normal Approximations to the Binomial 08:56
- S23.2 Poisson Arrivals During an Exponential Interval 09:37
- L24.1 Lecture Overview 01:59
- L24.2 Introduction to Markov Processes 02:09
- L24.3 Checkout Counter Example 12:10
- L24.4 Discrete-Time Finite-State Markov Chains 07:53
- L24.5 N-Step Transition Probabilities 10:59
- L24.6 A Numerical Example - Part I 09:26
- L24.7 Generic Convergence Questions 05:32
- L24.8 Recurrent and Transient States 05:37
- L25.1 Brief Introduction (RES.6-012 Introduction to Probability) 01:40
- L25.2 Lecture Overview 01:05
- L25.3 Markov Chain Review 06:15
- L25.4 The Probability of a Path 06:39
- L25.5 Recurrent and Transient States: Review 03:26
- L25.6 Periodic States 06:49
- L25.7 Steady-State Probabilities and Convergence 09:13
- L25.8 A Numerical Example - Part II 03:58
- L25.9 Visit Frequency Interpretation of Steady-State Probabilities 05:19
- L25.10 Birth-Death Processes - Part I 08:56
- L25.11 Birth-Death Processes - Part II 08:57
- L26.1 Brief Introduction (RES.6-012 Introduction to Probability) 01:41
- L26.2 Lecture Overview 00:40
- L26.3 Review of Steady-State Behavior 09:12
- L26.4 A Numerical Example - Part III 10:35
- L26.5 Design of a Phone System 18:30
- L26.6 Absorption Probabilities 09:58
- L26.7 Expected Time to Absorption 11:30
- L26.8 Mean First Passage Time 08:44
- L26.9 Gambler's Ruin 11:24
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