Probability Theory: Random Variables and Limit Theorems

Understanding how randomness behaves is a critical skill for anyone working with data, finance, or engineering. This course provides a structured path from basic probability concepts to the powerful theorems that govern statistical patterns in the real world. You will develop a deep intuition for how random processes stabilize over time and how to model uncertainty with precision. By the end of this course, you will be able to analyze complex random variables and apply the fundamental laws that underpin modern statistics. What you'll learn: - Define and classify various types of discrete and continuous random variables - Calculate expectation, variance, and higher-order moments to describe distributions - Apply the Law of Large Numbers to understand long-term stability in random events - Master the Central Limit Theorem to approximate sums of independent variables - Understand the foundations of Monte Carlo methods for numerical simulation - Practice solving rigorous probability problems using detailed written guides The course begins with essential terminology and the definition of random variables before moving into distribution functions and the transformative limit theorems used in modern data analysis. This program is designed for beginners with a basic grasp of mathematics who want to strengthen their theoretical and practical understanding of probability. Begin your journey into the mathematics of uncertainty today.