Foundations of Probability and Random Variables

Overview

Class Central Tips

The course "Foundations of Probability and Random Variables" introduces fundamental concepts in probability and random variables, essential for understanding computational methods in computer science and data science. Through five comprehensive modules, learners will explore combinatorial analysis, probability, conditional probability, and both discrete and continuous random variables. By mastering these topics, students will gain the ability to solve complex problems involving uncertainty, design probabilistic models, and apply these concepts in fields like machine learning, AI, and algorithm design. What makes this course unique is its practical approach: students will develop hands-on proficiency in the R programming language, which is widely used in data science and statistical modeling. The course also includes real-world applications, allowing learners to bridge theoretical knowledge with practical problem-solving skills. Whether you are aiming to pursue advanced studies in machine learning or develop data-driven solutions in professional settings, this course provides the solid foundation you need to excel. Designed for learners with a background in calculus and basic programming, this course prepares you to tackle more advanced topics in computational science.

Syllabus

Course Introduction

This course provides a comprehensive introduction to fundamental concepts in probability and statistics, focusing on counting principles, permutations, combinations, and multinomial coefficients. Students will explore probability axioms, conditional probabilities, and Bayes’s Formula while using Venn diagrams to visualize events. The course covers random variables, including discrete and continuous types, expected values, and various probability distributions. Practical applications in R programming and data analysis tools will enhance understanding through simulations and real-world problem-solving. By the end, students will be equipped to analyze and interpret statistical data effectively.

Combinatorial Analysis

This module covers the usefulness of an effective method for counting the number of ways that things can occur. Many problems in probability theory can be solved simply by counting the number of different ways that a certain event can occur.

Probability

This module introduces the concept of the probability of an event and then shows how probabilities can be computed in certain situations.

Conditional Probability and Independence

This module explores one of the most important concepts in probability theory, that of conditional probability. The importance of this concept is twofold. First, we are often interested in calculating probabilities when some partial information concerning the result of an experiment is available; in such a situation, the desired probabilities are conditional. Second, even when no partial information is available, conditional probabilities can often be used to compute the desired probabilities more easily.

Discrete Random Variables

This module discusses the function of outcomes rather than the actual outcomes themselves. In particular, we examine random variables that can take on at most a countable number of possible values. We call these types of variables, discrete random variables.

Continuous Random Variables

This module extends the concept of random variables where the outcomes cannot be counted. We explore probability density functions, cumulative distribution functions, the normal distribution and other common distributions.