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Intro - Syllabus

Introduction to Probability - The Science of Uncertainty - MITx - 6.041x

An introduction to probabilistic models, including random processes and the basic elements of statistical inference.

Topics​

  • The basic structure and elements of probabilistic models
  • Random variables, their distributions, means, and variances
  • Probabilistic calculations
  • Inference methods (Bayesian Inference)
  • Laws of large numbers and their applications
  • Random processes (Poisson processes and Markov chains)
  • multiple discrete or continuous random variables, expectations, and conditional distributions

Course Objectives​

  • Probabilistic concepts, and language
  • Common models
  • Mathematical tools
  • Intuition
  • Acquire working knowledge of the subject

Syllabus​

Unit 1: Probability models and axioms​

L1: Probability models and axioms

  • Sample Space
    • Sample Space
    • Probability Law
  • Probability Axioms
    • Nonnegativity
    • Normalization
    • (Finite) Additivity
  • Properties of Probability
  • Discrete example
  • Continuous example
  • Countable additivity Mathematical Background: Set, Sequences, Limits and Series, (un)countable sets
  • Sets
  • De Morgan's Law
  • Sequences and their limits
  • When does a sequence converge
  • Infinite series
  • Geometric series
  • Order of summation of series with multiple indices
  • Countable and uncountable sets
  • Set of real numbers is uncountable - Cantor's diagonalization argument

Solved problems​

  • The probability of difference of two sets
  • Genuises and chocolates
  • Uniform probabilities on a square
  • Bonferroni's inequality

Unit 2: Conditioning and independence​

L2: Conditioning and Bayes' rule

  • Conditional Probability
    • Multiplication Rule
    • Total Probability Theorem
    • Bayes' Rule
  • A die roll example
  • Conditional probabilities obey the same axioms
  • A radar example: models based on conditional probabilities and three basic tools
  • The multiplication rule
  • Total probability theorem
  • Bayes' rule L3: Independence
  • Coin tossing example
  • Independence of two events
  • Independence of event complements
  • Conditional independence
  • Independence vs conditional independence
  • Independence of a collection of events
  • Independence vs pairwise independence
  • Reliability
  • The King's sibling

Solved problems​

  • Conditional probability example
  • A chess tournament puzzle
  • A coin tossing puzzle
  • The Monty Hall Problem
  • A random walker
  • Communication over a noisy channel
  • Network reliability

Unit 3: Counting​

L4: Counting

  • The counting principle
  • Die roll example
  • Combinations
  • Binomial Probabilities
  • A coin tossing examples
  • Partitions
    • Multinomial Coefficient
    • Binomial Coefficient
  • Each person gets an ace

Solved Problems​

  • The birthday paradox
  • Rooks on a chessboard
  • Hypergeometric probabilities
  • Multinomial probabilities

Unit 4: Discrete random variables​

L5: Probability mass functions and expectations

  • Definition of random variables
  • Probablity mass functions
  • Bernoulli and indicator random variables
  • Uniform random variables
  • Binomial random variables
  • Geometric random variables
  • Expectation
  • Elementary properties of expectation
  • The expected value rule
  • Linearity of expectations L6: Variance; Conditioning on an event; Multiple r.v.'s
  • Variance
  • Variance of the Bernoulli and the uniform
  • Conditional PMFs and expectations given an event
  • Total expectation theorem
  • Geometric PMF, memorylessness, and expection
  • Joint PMFs and the expected value rule
  • Linearity of expectations and the mean of the binomial L7: Conditioning on a random variable; Independence of r.v.'s
  • Conditional PMFs
  • Conditional expectation and the total expectation theorem
  • Independece of random variables
  • Independence and expectations
  • Independence, variances, and the binomial variance
  • The hat problem

Unit 5: Continuous random variables​

L8: Probability density functions

Unit 6: Further topics on random variables​

L11: Derived distributions

Unit 7: Bayesian inference​

L14: Introduction to Bayesian inference

Unit 8: Limit theorems and classical statistics​

L18: Inequalities, convergence, and the Weak Law of Large Numbers L19: The Central Limit Theorem (CLT) L20: An introduction to classical statistics

Unit 9: Bernoulli and Poisson processes​

L21: The Bernoulli process L22: The Poisson process L23: More on the Poisson process

Unit 10: Markov chains​

L24: Finite-state Markov chains L25: Steady-state behavior of Markov chains L26: Absorption probabilities and expected time to absorption

Course - MITx: Probability - The Science of Uncertainty and Data | edX

Syllabus - https://courses.edx.org/courses/course-v1:MITx+6.041x_4+1T2017/course