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MITx: Probability - The Science of Uncertainty and Data

Build foundational knowledge of data science with this introduction to probabilistic models, including random processes and the basic elements of statistical inference -- Part of the MITx MicroMasters program in Statistics and Data Science.

16 weeks
10–14 hours per week
Instructor-paced
Instructor-led on a course schedule
Free
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Starts May 14, 2025
Ends Sep 3, 2025
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Probability - The Science of Uncertainty and Data

About this course

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The world is full of uncertainty: accidents, storms, unruly financial markets, noisy communications. The world is also full of data. Probabilistic modeling and the related field of statistical inference are the keys to analyzing data and making scientifically sound predictions.

Probabilistic models use the language of mathematics. But instead of relying on the traditional "theorem-proof" format, we develop the material in an intuitive -- but still rigorous and mathematically-precise -- manner. Furthermore, while the applications are multiple and evident, we emphasize the basic concepts and methodologies that are universally applicable.

The course covers all of the basic probability concepts, including:

  • multiple discrete or continuous random variables, expectations, and conditional distributions
  • laws of large numbers
  • the main tools of Bayesian inference methods
  • an introduction to random processes (Poisson processes and Markov chains)

The contents of this courseare heavily based upon the corresponding MIT class -- Introduction to Probability -- a course that has been offered and continuously refined over more than 50 years. It is a challenging class but will enable you to apply the tools of probability theory to real-world applications or to your research.

This course is part of theMITx MicroMasters Program in Statistics and Data Science. Master the skills needed to be an informed and effective practitioner of data science. You will complete this course and three others from MITx, at a similar pace and level of rigor as an on-campus course at MIT, and then take a virtually-proctored exam to earn your MicroMasters, an academic credential that will demonstrate your proficiency in data science or accelerate your path towards an MIT PhD or a Master's at other universities. To learn more about this program, please visit https://micromasters.mit.edu/ds/.

At a glance

  • Associated skills:Data Science, Stochastic Process, Data Analysis, Probability Theories, Markov Chain, Statistics, Financial Market, Random Variables, Communications, Probability, Statistical Inference, Bayesian Inference, Basic Math

What you'll learn

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  • The basic structure and elements of probabilistic models
  • Random variables, their distributions, means, and variances
  • Probabilistic calculations
  • Inference methods
  • Laws of large numbers and their applications
  • Random processes

Syllabus

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Unit 1: Probability models and axioms

  • Probability models and axioms
  • Mathematical background: Sets; sequences, limits, and series; (un)countable sets.

Unit 2: Conditioning and independence

  • Conditioning and Bayes' rule
  • Independence

Unit 3: Counting

  • Counting

Unit 4: Discrete random variables

  • Probability mass functions and expectations
  • Variance; Conditioning on an event; Multiple random variables
  • Conditioning on a random variable; Independence of random variables

Unit 5: Continuous random variables

  • Probability density functions
  • Conditioning on an event; Multiple random variables
  • Conditioning on a random variable; Independence; Bayes' rule

Unit 6: Further topics on random variables

  • Derived distributions
  • Sums of independent random variables; Covariance and correlation
  • Conditional expectation and variance revisited; Sum of a random number of independent random variables

Unit 7: Bayesian inference

  • Introduction to Bayesian inference
  • Linear models with normal noise
  • Least mean squares (LMS) estimation
  • Linear least mean squares (LLMS) estimation

Unit 8: Limit theorems and classical statistics

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

Unit 9: Bernoulli and Poisson processes

  • The Bernoulli process
  • The Poisson process
  • More on the Poisson process

Unit 10 (Optional): Markov chains

  • Finite-state Markov chains
  • Steady-state behavior of Markov chains
  • Absorption probabilities and expected time to absorption

Learner testimonials

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“This is by far the best probability & statistics course available--online or in the classroom.”

"You won’t find another intro to probability with greater depth and breadth."

"This is a great course for those serious about forming a solid foundation in probability."

"[This course] has created a love for probabilistic models, that, I guess, truly govern everything around us."

"This should be in top 10 MOOCs of all time."

Frequently Asked Questions

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How is this class related to 6.041x?
The material covered, and the resources (videos, etc.) are largely the same, but homeworks and exams contain revised and new problems.

What textbook do I need for the course?
None - there is no required textbook. The class follows closely the text I ntroduction to Probability, 2nd edition, by Bertsekas and Tsitsiklis, Athena Scientific, 2008. (See the publisher's website or Amazon.com for more information.) However, while this textbook is recommended as supplemental reading, the materials provided by this course are self-contained.

What is the format of the class?
The course material is organized along units, each unit containing between one and three lecture sequences. (For those who purchase the textbook, each unit corresponds to a chapter.) Each lecture sequence consists of short video clips,interwovenwith short problems to test your understanding. Each unit also contains a wealth of supplementary material, including videos that go through the solutions to various problems.

How much do I need to work for this class?
This is an ambitious class in that it covers a lot of material in substantial depth. In addition, MIT considers that the best way to master the subject is by actually solving on your own a fair number of problems. MIT students who take the corresponding residential class typically report an average of 11-12 hours spent each week, including lectures, recitations, readings, homework, and exams.

More about this course

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For more information, please visit https://micromasters.mit.edu/ds/faq/.

Who can take this course?

Unfortunately, learners residing in one or more of the following countries or regions will not be able to register for this course: Iran, Cuba and the Crimea region of Ukraine. While edX has sought licenses from the U.S. Office of Foreign Assets Control (OFAC) to offer our courses to learners in these countries and regions, the licenses we have received are not broad enough to allow us to offer this course in all locations. edX truly regrets that U.S. sanctions prevent us from offering all of our courses to everyone, no matter where they live.

This course is part of Statistics and Data Science (Time Series and Social Sciences Track) MicroMasters Program

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Expert instruction
5 graduate-level courses
Instructor-led
Assignments and exams have specific due dates
1 year 1 month
10 - 14 hours per week

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