Discrete Mathematics

1. Discrete Probability
   1. Basic Definitions
      1. Probability Spaces
         1. Experiments
            1. Definition of an Experiment
            2. Examples of Discrete Experiments
         2. Sample Space
            1. Definition and Examples
            2. Finite vs Infinite Sample Spaces
         3. Events
            1. Definition of Events
            2. Types of Events: Simple and Compound
            3. Equal likelihood of outcomes
      2. Probability Axioms
         1. Axiom of Non-Negativity
         2. Axiom of Normalization
         3. Axiom of Additivity
      3. Conditional Probability
         1. Definition and Interpretation
         2. Properties of Conditional Probability
            1. Multiplicative Rule for Probability
            2. Bayes' Theorem
            3. Examples and Applications
      4. Independence
         1. Definition of Independence between Events
         2. Properties and Examples of Independent Events
         3. Testing Independence
         4. Mutual Independence vs Pairwise Independence
   2. Random Variables
      1. Discrete Random Variables
         1. Definition and Examples of Discrete Random Variables
         2. Difference between Discrete and Continuous Random Variables
      2. Probability Mass Function (PMF)
         1. Definition and Properties of PMF
         2. Calculation of PMF for Different Random Variables
         3. Relationship between PMF and Distribution Function
      3. Cumulative Distribution Function (CDF)
         1. Definition and Properties of CDF
         2. Steps to derive CDF from PMF
         3. Piecewise Representation of CDF
      4. Expected Value and Variance
         1. Definition and Computation of Expected Value
         2. Properties of Expectation
            1. Linearity of Expectation
            2. Expected Value of Sums and Multiplied Constants
         3. Definition and Interpretation of Variance
         4. Calculation of Variance and Standard Deviation
         5. Properties of Variance
         6. Chebyshev's Inequality
   3. Common Discrete Distributions
      1. Bernoulli Distribution
         1. Definition and PMF
         2. Expected Value and Variance
      2. Binomial Distribution
         1. Definition and Characteristics
         2. Binomial Coefficients in PMF
         3. Mean and Variance of Binomial Distribution
         4. Applications and Examples
      3. Geometric Distribution
         1. Definition and Properties
         2. Memoryless Property
         3. Expected Value and Variance
      4. Poisson Distribution
         1. Definition and PMF
         2. Limiting Case of Binomial Distribution
         3. Mean and Variance of Poisson Distribution
         4. Use in Modeling Count Data
   4. Advanced Topics in Discrete Probability
      1. Joint Distribution of Discrete Random Variables
         1. Joint PMF and Marginal PMF
         2. Covariance and Correlation
         3. Independence of Random Variables
      2. Moment Generating Functions
         1. Definition and Properties
         2. Deriving Mean and Variance from MGF
      3. Probability Inequalities
         1. Markov's Inequality
         2. Hoeffding's Inequality
      4. Law of Large Numbers
         1. Weak Law of Large Numbers
         2. Strong Law of Large Numbers
      5. Central Limit Theorem for Discrete Distributions
         1. Conditions and Applications
         2. Approximating Binomial and other Distributions