Probability - The Science of Uncertainty and Data

Description

In this course, you will learn:

• Probabilistic models' essential structure and elements.
• The distributions, means, and variances of random variables.
• Calculations based on probabilities.
• Methods of inference
• Large-number laws and their applications
• Processes that happen at random

Syllabus:

1. Probability models and axioms

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

2. Conditioning and independence

• Conditioning and Bayes' rule
• Independence
   

3. Counting

• Counting
   

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
   

5. Continuous random variables

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

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
   

7. Bayesian inference

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

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
   

9. Bernoulli and Poisson processes

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

10. Markov chains

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

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