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# Probability - The Science of Uncertainty and Data

### Overview

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/.

### Syllabus

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

### Taught by

Patrick Jaillet and John Tsitsiklis

## Reviews

4.9 rating, based on 34 reviews

Start your review of Probability - The Science of Uncertainty and Data

• Dolly Ye

Dolly Ye is taking this course right now, spending 14 hours a week on it and found the course difficulty to be hard.

This course is very thorough and challenging.The learning curve could be steep at times,but don't get frustrated. The problem sets are structured to deepen your understanding of the materials.I partially completed it a year before,but taking it again...
• Gregory J Hamel ( Life Is Study) completed this course and found the course difficulty to be hard.

6.041x: Introduction to Probability - The Science of Uncertainty is a comprehensive 16-week introduction to probability offered by MIT through the edX MOOC platform. Although this course is dubbed an “introduction” it is not easy. You need familiarity...
• Anonymous
Very good course. Can be challenging at times but I definitely learnt a lot. everything is well put together, recitations are on point, just very good.

Not 5 stars because we had some discussions about some answers to the final exam, the staff told us they would come back to us to answer but instead waited until the forums were frozen at the end of the course so that they didn't have to answer. This was the only hiccup (and in the end doesn't matter that much), the whole rest of the course was excellent from start to almost finish.
• Bart

Bart completed this course.

The best courses I have taken. I have some prior experience but that is very rusty and definitely not as extensive as the topics in this course. The professor has a very clear way of explaining the topics. There is hardly any repetition and a good balance...
• Ilir Sheraj
In my opinion this course is one of the top-5 best courses on MOOC. Its is not a wash-out course for mass consumption or cheap way to receive a certificate, but it is really challenging and requires a lot of time to follow the lectures, read the textbook,...
• What to say? It is simply THE best MOOC you can find, given that you have the time to study the subject. You need neither the interest nor good knowledge on the subject: the interest will just come by attending the course teach by these superb professors,...
• HChan

HChan completed this course, spending 8 hours a week on it and found the course difficulty to be medium.

Many online courses are watered down in some way, but this one feels like a proper rigorous exercise-driven course similar to what you'd get in-person at a top school like MIT. The professors present concepts in lectures that have obviously been honed to a laser focus through years of pedogogical experience - there is not a single wasted second in the presentations and they go exactly at the right pace and detail for you to understand the concepts. The exercises will make you work for your knowledge and are critical for really internalizing the concepts. This is the best online course I have taken in any subject.
• Siddharth Gupta completed this course, spending 14 hours a week on it and found the course difficulty to be medium.

This is the most beautifully designed course I have ever attended in my life, having completed first 4 weeks, it has been both rigorous and to the point. This course questions the whole pedagogy that I have faced in India. If someone fails to attempt the questions after the lectures, it is because of his inefficiency of grasping the concepts, the questions are designed in such a way that they would test your learning to the core.
• Jiting Tian completed this course, spending 16 hours a week on it and found the course difficulty to be hard.

This is an introductory course on probability theory, but, it's very hard (after all, it's from MIT). The materials, which have covered all the related topics on probability, are organized quite well and illustrated in a gradual and clear way. A lot of difficult exercises are required, but they are very useful to help students understand the concepts and master the calculation ways. The whole course lasts for 16 weeks (oh my god!), but when I insist on to the end, I have learnt so much and feel so satisfied. Thank you, Prof. John Tsitsiklis and the course staff!
• Nerpatu
- A fun course if you like math.
- The solution largely depends upon some intuition without proofs, as in other MIT courses.
- You will get the practical skills.
- This course is not for a math major.
- The contents do not change over the years. No new materials.
• Anonymous
Excellent probability introduction. Clear explanation of theory and representative examples presented by a top scientist. I believe this is the best online probability course available.
• Anonymous

Anonymous is taking this course right now.

The style of this course is to start from complete basics and then slowly building complexity. This is often what courses try to do, but they rarely succeed this well in my opinion. I feel like I have a solid grasp on the topics that I've completed thus far.

Thanks to EdX for providing this amazing free resource!
• Federico Carrone

Federico Carrone is taking this course right now, spending 12 hours a week on it and found the course difficulty to be hard.

It is an introductory course but, I would not recommend it to somebody that doesn't have any idea of probability or statistics. The problems and exams have some difficult exercises. After taking many probability courses, this is the first time I feel I really understand it. It is fun and rigorous at the same. It is one of the best MOOC I've taken if not the best.
• Federico Carrone

Federico Carrone is taking this course right now, spending 12 hours a week on it and found the course difficulty to be hard.

This is a great introducing course on probability. A certain level in math is a prerequisite, but nothing complicated. The teacher is clear and the his explanations really help to understand notion that can appear complicated at first glance. The exercices are designed to help the understanding. They're not "challenging", but are helpful.
• Anonymous

Anonymous is taking this course right now.

This course is just perfect! One of the best moocs you can find ever! It covers a lot, and it's rigorous and demanding. But everything is explained very clearly and the course team help a lot. Thanks for the team and hope there would be more courses coming from the team.
• Anonymous

Anonymous completed this course.

This course is really a legend! It's very challenging but professor explained all the things so well and smoothly. I have taken a course in probability and found it so hard that I cannot understand a lot of theory. But through this course, I really understand a lot.
• Anonymous

Anonymous is taking this course right now.

I am taking a graduate version of the course numbered MITx 6.431 to end at the end of Dec'18. It is the best MOOC I have ever taken. In fact, given what is generally thought of MOOC in terms of rigour, and level of difficulty, this course is nothing like...
• Jinqiang Zhang

Jinqiang Zhang completed this course, spending 10 hours a week on it and found the course difficulty to be very hard.

I tried 2 or 3 times for the course, it's very hard. It's hard because it has more content than a usual probability course. The professor is very good, nice accent, smart guy.

It's good to have some calculus knowledge prior to this course, because you don't want to handle the difficulty from the course itself, as well as the technique issue from calculus. At the end of the course, the classical part of the probability is quite different from the front parts, I don't feel I had a firm grasp of the ideas, I guess at some point I need to revisit this part.