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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 course are 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 the MicroMasters Program in Statistics and Data Science.

**What you'll learn:**

- The basic structure and elements of probabilstic 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

- 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

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