Module 1: Theoretical Foundations
1. Introduction to Probability Theory
1.1 Basic Concepts
- Events and Sample Spaces
- Definition and examples of events
- Types of sample spaces (finite, countable, uncountable)
- Set operations and their implications on events (union, intersection, complement)
- Probability Measures
- Definition of probability measure
- Examples of probability measures in different sample spaces
1.2 Properties of Probability Measures
- Additivity
- Finite additivity: Probability of the union of two disjoint events
- Countable Additivity
- Definition and implications for infinite sample spaces
- Sigma-algebras and their role in defining probability measures
1.3 Probability Axioms
- Axioms of Probability
- Non-negativity: ( P(A) \geq 0 )
- Normalization: ( P(S) = 1 )
- Additivity: ( P(A \cup B) = P(A) + P(B) ) for disjoint ( A ) and ( B )
- Implications and Applications
- Examples illustrating the axioms
- Practical applications in real-world scenarios
2. Conditional Probability and Independence
2.1 Conditional Probability
- Definition and Notation
- ( P(A|B) = \frac{P(A \cap B)}{P(B)} ), provided ( P(B) > 0 )
- Properties and Examples
- Law of total probability
- Bayes’ theorem and its applications
2.2 Independence
- Definition of Independent Events
- Two events ( A ) and ( B ) are independent if ( P(A \cap B) = P(A)P(B) )
- Properties of Independence
- Pairwise independence vs. mutual independence
- Examples and counterexamples
3. Random Variables and Probability Distributions
3.1 Random Variables
- Definition and Types
- Discrete and continuous random variables
- Examples of each type
- Functions of Random Variables
- Transformation of random variables
- Examples of common transformations
3.2 Probability Distributions
- Discrete Distributions
- Probability mass functions (PMFs)
- Common discrete distributions (Bernoulli, Binomial, Poisson)
- Continuous Distributions
- Probability density functions (PDFs)
- Common continuous distributions (Uniform, Normal, Exponential)
- Cumulative Distribution Functions (CDFs)
- Definition and properties
- Relation between PDF/PMF and CDF
By the end of this module, students should have a strong foundation in the basic principles of probability theory, be able to work with conditional probabilities and independence, and understand the concept and properties of random variables and their associated probability distributions.