Probability - The Science of Uncertainty and Data from edX

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/.

What's inside

Learning objectives

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

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

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

Highly rated probability course

Learners say this probability course is a top-rated offering with challenging but engaging assignments. The course is highly rigorous and demanding but provides a solid mathematical foundation for understanding the fundamentals of probability. Instructors present clear lectures and explanations, making complex concepts more approachable. The course team and forum discussions are also quite helpful. Overall, this course is highly recommended for anyone interested in probability, whether for academic or professional pursuits.

The course team and forum discussions provide helpful support to students.

"The course team help a lot."

"Many thanks to the teaching teams who were always helpful and the other students for their great discussions in the forum."

Instructors present clear lectures and explanations, making complex probability concepts more understandable.

"Clear explanation of theory and representative examples presented by a top scientist."

"The teacher is clear and the his explanations really help to understand notion that can appear complicated at first glance."

The course offers challenging and engaging assignments that help students develop practical skills.

"You will get the practical skills."

"The exercices are designed to help the understanding."

"After taking many probability courses, this is the first time I feel I really understand it. It is fun and rigorous at the same."

The course covers a lot of material and is quite rigorous and demanding.

"This course is just perfect! One of the best moocs you can find ever! It covers a lot, and it's rigorous and demanding."

"After completed all the course of the MicroMaster, and now reviewing for the final exam, my rate didn't change."

The course may not be suitable for beginners who have no prior knowledge of probability or statistics.

"I would not recommend it to somebody that doesn't have any idea of probability or statistics."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Probability - The Science of Uncertainty and Data with these activities:

Review Probability and Statistics Textbook

Show steps

Strengthen your foundation by reviewing a comprehensive probability and statistics textbook.

View Probability & Statistics for Engineers &... on Amazon

Show steps

Read through the relevant chapters, focusing on core concepts and examples
Solve practice problems to test your understanding
Highlight and summarize important points for easy reference

Explore Online Probability Tutorials

Show steps

Expand your knowledge and refine your skills by exploring online resources and tutorials.

Browse courses on Online Learning

Show steps

Identify credible online sources for probability tutorials
Follow along with tutorials, taking notes and practicing problems
Seek support from online forums or communities if needed

Review Topics in Probability & Statistics

Show steps

Refresh your understanding of probability and statistics concepts as they relate to this course.

Browse courses on Probability

Show steps

Revisit probability axioms and concepts
Review key probability distributions, such as binomial, normal, and Poisson
Brush up on statistical inference methods, including hypothesis testing and confidence intervals

Six other activities

Expand to see all activities and additional details

Show all nine activities

Participate in Probability Study Groups

Show steps

Engage with other learners to discuss concepts, clarify doubts, and reinforce your understanding.

Browse courses on Collaboration

Show steps

Find or form a study group with peers
Regularly meet to discuss course material, solve problems, and share insights
Take turns presenting different topics and facilitating discussions

Work Through Probability Problems

Show steps

Solidify your understanding of probabilistic concepts and develop problem-solving skills.

Browse courses on Problem Solving

Show steps

Identify the relevant concepts and variables
Set up and solve probability problems using appropriate distributions
Interpret and draw conclusions from the results

Participate in Probability Challenges

Show steps

Test your skills and expand your knowledge by participating in probability-related challenges.

Browse courses on Challenges

Show steps

Find and register for relevant challenges or competitions
Analyze the problem statements and develop innovative solutions
Submit your solutions and receive feedback

Create Probability Explanation Videos

Show steps

Enhance your understanding by creating videos that explain probability concepts in a clear and engaging manner.

Show steps

Choose a specific probability topic to cover
Gather relevant information and present it in a logical and visually appealing way
Record and edit the video, paying attention to clarity and engagement
Share your videos with others for feedback

Develop a Probability Application Project

Show steps

Apply your knowledge by creating a project that demonstrates the practical applications of probability.

Browse courses on Project-Based Learning

Show steps

Identify a real-world problem or scenario that can be analyzed using probability
Develop a project plan, outlining the problem statement, data collection methods, and analysis techniques
Collect and analyze data using probabilistic models
Present your findings and conclusions in a clear and concise manner

Build a Probability Simulator

Show steps

Deepen your understanding of probability by creating a simulator that models different scenarios.

Browse courses on Programming

Show steps

Design and implement a simulation algorithm
Test and refine your simulator to ensure accurate results
Use the simulator to explore probability distributions, laws of large numbers, and other concepts

Career center

Learners who complete Probability - The Science of Uncertainty and Data will develop knowledge and skills that may be useful to these careers:

Statistician

A Statistician collects, analyzes, interprets, and presents data to provide insights for decision-making. This course, Probability - The Science of Uncertainty and Data, provides a solid foundation in probability and statistics, which are essential for understanding the role of data in decision-making. The course covers topics such as probability distributions, random variables, statistical inference, and regression analysis, which are all highly relevant to the field of statistics.

See salaries and explore the career path for Statistician

Data Scientist

A Data Scientist extracts knowledge and insights from data by applying statistical and machine learning techniques. This course, Probability - The Science of Uncertainty and Data, provides a solid foundation in probability and statistics, which are essential for understanding the role of data in machine learning and artificial intelligence. The course covers topics such as probability distributions, random variables, statistical inference, and regression analysis, which are all highly relevant to data science.

See salaries and explore the career path for Data Scientist

Machine Learning Engineer

A Machine Learning Engineer designs, develops, and deploys machine learning models to solve real-world problems. This course, Probability - The Science of Uncertainty and Data, provides a solid foundation in probability and statistics, which are essential for understanding the role of probability and randomness in machine learning. The course covers topics such as probability distributions, random variables, statistical inference, and regression analysis, which are all highly relevant to machine learning engineering.

See salaries and explore the career path for Machine Learning Engineer

Risk Manager

A Risk Manager identifies, assesses, and mitigates risks to an organization. This course, Probability - The Science of Uncertainty and Data, is designed for those interested in gaining a deep understanding of probability and statistics, which are essential for understanding risk assessment and management. The course covers topics such as probability distributions, random variables, and statistical inference, which are all highly relevant to risk management.

See salaries and explore the career path for Risk Manager

Actuary

An Actuary analyzes and manages financial risks for insurance companies and other financial institutions. The course, Probability - The Science of Uncertainty and Data, can be particularly useful for those interested in gaining a strong foundation in probability and statistics, which are essential for understanding actuarial science. The course covers topics such as probability distributions, random variables, and statistical inference, which are all highly relevant to actuarial work.

See salaries and explore the career path for Actuary

Quantitative Analyst

A Quantitative Analyst develops and uses mathematical and statistical models to analyze financial data and make investment decisions. This course, Probability - The Science of Uncertainty and Data, can be particularly useful for those interested in building a strong foundation in probability and statistics, which are essential for understanding financial modeling and risk assessment. The course covers topics such as probability distributions, random variables, and statistical inference, which are all highly relevant to quantitative analysis.

See salaries and explore the career path for Quantitative Analyst

Biostatistician

A Biostatistician applies statistical methods to solve problems in biology and medicine. The course, Probability - The Science of Uncertainty and Data, can be particularly useful for those interested in gaining a strong foundation in probability and statistics, which are essential for understanding biostatistical research. The course covers topics such as probability distributions, random variables, and statistical inference, which are all highly relevant to biostatistics.

See salaries and explore the career path for Biostatistician

Epidemiologist

An Epidemiologist investigates the causes and patterns of health and disease in populations. The course, Probability - The Science of Uncertainty and Data, can be particularly useful for those interested in gaining a strong foundation in probability and statistics, which are essential for understanding epidemiological research. The course covers topics such as probability distributions, random variables, and statistical inference, which are all highly relevant to epidemiology.

See salaries and explore the career path for Epidemiologist

Financial Analyst

A Financial Analyst analyzes investment opportunities and provides recommendations to clients. This course, Probability - The Science of Uncertainty and Data, can be particularly useful for those interested in building a solid foundation in probability and statistics, which are essential for understanding financial modeling and risk assessment. The course covers topics such as probability distributions, random variables, and statistical inference, which are all highly relevant to financial analysis.

See salaries and explore the career path for Financial Analyst

Market Researcher

A Market Researcher collects and analyzes data about markets, customers, and competitors. The course, Probability - The Science of Uncertainty and Data, can be helpful for those interested in gaining a strong foundation in probability and statistics, which are essential for understanding data collection and analysis techniques. The course covers topics such as probability distributions, random variables, and statistical inference, which are all highly relevant to market research.

See salaries and explore the career path for Market Researcher

Data Analyst

A Data Analyst typically analyzes a large amount of data by applying statistical methods and techniques. This course, Probability - The Science of Uncertainty and Data, may be useful because it covers several topics that are relevant to data science, including probability models, random variables, and statistical inference. This course can be particularly relevant to those who wish to gain a solid understanding of the foundational concepts of probability and statistics, which are essential for many data analyst roles.

See salaries and explore the career path for Data Analyst

Operations Research Analyst

An Operations Research Analyst uses mathematical and statistical techniques to solve complex problems in business and industry. This course, Probability - The Science of Uncertainty and Data, may be useful because it covers several topics that are relevant to operations research, including probability models, optimization, and decision theory.

See salaries and explore the career path for Operations Research Analyst

Operations Manager

An Operations Manager plans and oversees the production and delivery of goods and services. The course, Probability - The Science of Uncertainty and Data, may be useful because it covers several topics that are relevant to operations management, including probability models, random variables, and statistical inference.

See salaries and explore the career path for Operations Manager

Economist

An Economist studies the production, distribution, and consumption of goods and services. The course, Probability - The Science of Uncertainty and Data, may be useful because it covers several topics that are relevant to economics, including probability models, random variables, and statistical inference.

See salaries and explore the career path for Economist

Software Engineer

A Software Engineer designs, develops, and maintains software applications. This course, Probability - The Science of Uncertainty and Data, may be useful because it covers several topics that are relevant to software engineering, including probability models, random variables, and statistical inference.

See salaries and explore the career path for Software Engineer