Introduction to Probability
MIT RES.6-012, Spring 2018 â Prof. John Tsitsiklis
A comprehensive course covering probability models, random variables, distributions, limit theorems, and stochastic processes. 100 video segments organized by lecture topic.
Lecture 1: Probability Models and Axioms
Sample spaces, probability axioms, discrete and continuous examples, countable additivity.
L01.1 Lecture Overview
L01.2 Sample Space
L01.3 Sample Space Examples
L01.4 Probability Axioms
L01.5 Simple Properties of Probabilities
L01.6 More Properties of Probabilities
L01.7 A Discrete Example
L01.8 A Continuous Example
L01.9 Countable Additivity
L01.10 Interpretations & Uses of Probabilities
Supplement 1: Mathematical Background
Sets, De Morgan's laws, sequences, series, countable and uncountable sets.
S01.0 Mathematical Background Overview
S01.1 Sets
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
S01.10 Bonferroni's Inequality
Lecture 2: Conditioning and Bayes' Rule
Conditional probabilities, the multiplication rule, total probability theorem, and Bayes' rule.
L02.1 Lecture Overview
L02.2 Conditional Probabilities
L02.3 A Die Roll Example
L02.4 Conditional Probabilities Obey the Same Axioms
L02.5 A Radar Example and Three Basic Tools
L02.6 The Multiplication Rule
L02.7 Total Probability Theorem
L02.8 Bayes' Rule
Lecture 3: Independence
Independence of events, conditional independence, collections of events, pairwise independence, reliability.
L03.1 Lecture Overview
L03.2 A Coin Tossing Example
L03.3 Independence of Two Events
L03.4 Independence of Event Complements
L03.5 Conditional Independence
L03.6 Independence Versus Conditional Independence
L03.7 Independence of a Collection of Events
L03.8 Independence Versus Pairwise Independence
L03.9 Reliability
L03.10 The King's Sibling
Lecture 4: Counting
The counting principle, combinations, binomial and multinomial probabilities, partitions.
L04.1 Lecture Overview
L04.2 The Counting Principle
L04.3 Die Roll Example
L04.4 Combinations
L04.5 Binomial Probabilities
L04.6 A Coin Tossing Example
L04.7 Partitions
L04.8 Each Person Gets An Ace
L04.9 Multinomial Probabilities
Lecture 5: Discrete Random Variables
Definition of random variables, PMFs, Bernoulli, uniform, binomial, geometric distributions, expectation.
L05.1 Lecture Overview
L05.2 Definition of Random Variables
L05.3 Probability Mass Functions
L05.4 Bernoulli & Indicator Random Variables
L05.5 Uniform Random Variables
L05.6 Binomial Random Variables
L05.7 Geometric Random Variables
L05.8 Expectation
L05.9 Elementary Properties of Expectation
L05.10 The Expected Value Rule
L05.11 Linearity of Expectations
S05.1 Supplement: Functions
Lecture 6: Variance; Conditioning on an Event; Multiple r.v.'s
Variance, conditional PMFs and expectations, total expectation theorem, joint PMFs.
L06.1 Lecture Overview
L06.2 Variance
L06.3 The Variance of the Bernoulli & The Uniform
L06.4 Conditional PMFs & Expectations Given an Event
L06.5 Total Expectation Theorem
L06.6 Geometric PMF Memorylessness & Expectation
L06.7 Joint PMFs and the Expected Value Rule
L06.8 Linearity of Expectations & The Mean of the Binomial
Lecture 7: Conditioning on a Random Variable; Independence
Conditional PMFs, conditional expectation, independence of random variables, the hat problem.
L07.1 Lecture Overview
L07.2 Conditional PMFs
L07.3 Conditional Expectation & the Total Expectation Theorem
L07.4 Independence of Random Variables
L07.5 Example
L07.6 Independence & Expectations
L07.7 Independence, Variances & the Binomial Variance
L07.8 The Hat Problem
S07.1 The Inclusion-Exclusion Formula
S07.2 The Variance of the Geometric
S07.3 Independence of Random Variables Versus Independence of Events
Lecture 8: Continuous Random Variables
Probability density functions, uniform, exponential, and normal distributions, CDFs.
L08.1 Lecture Overview
L08.2 Probability Density Functions
L08.3 Uniform & Piecewise Constant PDFs
L08.4 Means & Variances
L08.5 Mean & Variance of the Uniform
L08.6 Exponential Random Variables
L08.7 Cumulative Distribution Functions
L08.8 Normal Random Variables
L08.9 Calculation of Normal Probabilities
Lecture 9: Multiple Continuous Random Variables
Conditioning on events and random variables, memorylessness, mixed r.v.'s, joint PDFs and CDFs.
L09.1 Lecture Overview
L09.2 Conditioning A Continuous Random Variable on an Event
L09.3 Conditioning Example
L09.4 Memorylessness of the Exponential PDF
L09.5 Total Probability & Expectation Theorems
L09.6 Mixed Random Variables
L09.7 Joint PDFs
L09.8 From The Joint to the Marginal
L09.9 Continuous Analogs of Various Properties
L09.10 Joint CDFs
S09.1 Buffon's Needle & Monte Carlo Simulation
Lecture 10: Continuous Bayes' Rule; Derived Distributions
Continuous Bayes' rule, derived distributions, and functions of random variables.
L10.1 Lecture Overview