Foundations of Probability and Mathematical Statistics

Preparing for rigorous mathematical statistics exams requires a deep, conceptual understanding of probability theory and statistical inference. This text-based course guides you step-by-step through the complex mathematical frameworks without overwhelming jargon. You will transition from basic probability concepts to advanced statistical methodologies, building the analytical skills needed to solve challenging exam-style problems with confidence. What you'll learn: - Understand fundamental probability concepts, including conditional probability, Bayes' theorem, and independent events - Analyze discrete and continuous random variables, cumulative distribution functions, and expectation - Master standard probability distributions such as Binomial, Poisson, Normal, and Exponential - Apply limit theorems, including the Law of Large Numbers and the Central Limit Theorem, to statistical problems - Explore estimation theory, confidence intervals, and hypothesis testing frameworks - Practice solving rigorous mathematical derivations and numerical problems through structured written exercises The course begins with foundational definitions of probability before advancing systematically through random variables, joint distributions, and key statistical inference techniques. Each module provides detailed written proofs, step-by-step derivations, and practice problems to solidify your understanding. This course is designed for university students, graduate school aspirants, and anyone preparing for competitive exams in mathematics or statistics. No advanced prior knowledge of statistics is required, though a basic background in calculus is recommended. Start reading today to master the mathematical foundations of probability and statistics.