Probability & Statistics

Syllabus

1. Module 1

   Probability Foundations

   • Sample Spaces, Events, and Probability Measures
   • Counting-Based Probability
   • Conditional Probability
   • Independence and Dependence

2. Module 2

   Bayesian Reasoning

   • Law of Total Probability
   • Bayes’ Theorem
   • Base Rates and Diagnostic Testing
   • Odds, Likelihood Ratios, and Sequential Updating

3. Module 3

   Discrete Random Variables

   • Probability Mass Functions
   • Bernoulli and Binomial Distributions
   • Geometric and Negative-Binomial Models
   • Poisson Distribution and Rare Events

4. Module 4

   Continuous Random Variables

   • Density and Distribution Functions
   • Uniform and Exponential Distributions
   • Normal Distribution
   • Transformations of Random Variables

5. Module 5

   Expectation and Variability

   • Expected Value and Linearity
   • Variance and Standard Deviation
   • Covariance and Correlation
   • Conditional Expectation

6. Module 6

   Joint Distributions

   • Joint, Marginal, and Conditional Distributions
   • Independence of Random Variables
   • Sums of Random Variables
   • Multivariate Normal Models

7. Module 7

   Limit Theorems and Simulation

   • Law of Large Numbers
   • Central Limit Theorem
   • Monte Carlo Simulation
   • Concentration and Tail Bounds

8. Module 8

   Statistical Estimation

   • Populations, Samples, and Statistics
   • Point Estimation and Bias
   • Maximum Likelihood Estimation
   • Confidence Intervals

9. Module 9

   Hypothesis Testing

   • Null and Alternative Hypotheses
   • Test Statistics and p-Values
   • Type I and Type II Errors
   • Power, Multiple Testing, and Practical Significance

10. Module 10

    Regression and Bayesian Statistics

    • Simple Linear Regression
    • Multiple Regression and Diagnostics
    • Bayesian Priors and Posteriors
    • Posterior Prediction and Model Checking