Introduction to probability (Course & exercises)
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Date
2025-12-16
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ummto.faculté des sciences
Abstract
This manual has been carefully structured to guide you from fundamental concepts to more advanced topics, ensuring a solid and progressive understanding.The journey begins with Chapter 1, which serves as a crucial recap of fundamental probability notions.Here, we will revisit the axiomatic foundations of probability, combinatorial calculations, conditional probability, and Bayes’ Theorem. This chapter is designed to solidify your intuition and ensure all students have a common, robust starting point. Numerous examples and solved exercises will help you master these essential tools. In Chapter 2, we will move from abstract events to more manageable mathematical
objects by introducing Random Variables. We will explore the key distinctions between discrete and continuous random variables, focusing on their characterization through probability mass functions, probability density functions, and cumulative distribution functions. This chapter will also introduce you to the important concepts of expectation and variance, which capture the ”average” behaviour and the spread of a random variable, respectively.
Chapter 3 builds directly upon this foundation by delving into the most Common
Probability Distributions. You will become acquainted with the essential families of discrete distributions (such as the Binomial, Poisson, and Geometric) and continuous distributions (such as the Uniform, Normal, and Exponential). A significant part of this chapter is dedicated to approximations, particularly the use of the Poisson distribution to approximate the Binomial, a powerful technique for simplifying complex calculations. Furthermore, we will explore the transformation of random variables, a key concept for understanding how distributions change under functional mappings, which is vital for more advanced statistical inference.
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Keywords
Probability, Combinatorial calculus, Bayes' Theorem, Random variables, Mass function, Binomial law, Statistical inference