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