Mathematics of Uncertainty

This comprehensive course on Probability and Statistics integrates theory with hands-on Python programming. Topics cover descriptive methods, probability concepts, random variables, statistical distributions, joint densities, sampling methods, and the Central Limit Theorem. Advanced modules address hypothesis testing, p-values, and statistical inference. Lab sessions build practical skills in data visualization, simulation, and implementation of probability rules. Students leave equipped to model uncertainty and draw informed conclusions from data in diverse contexts.

This course provides a comprehensive introduction to Probability and Statistics, integrating theoretical concepts with hands-on computational practice. Over 15 weeks, students will explore statistical methods, probability foundations, and random variables, transitioning from descriptive analysis to hypothesis testing. Python is used extensively for data visualization, simulation, and statistical computation. Key topics include probability distributions, the Central Limit Theorem, sampling, and advanced hypothesis testing. Laboratory sessions focus on practical applications, such as simulating distributions, computing statistical measures, and exploring real-world datasets. By the end of the course, students will have a robust understanding of statistical principles and their applications in data analysis.

By the end of this course, students will have had the opportunity to:

• Understand Statistical and Probabilistic Foundations: 
  Gain a comprehensive understanding of basic statistical concepts, including descriptive methods, probability axioms, and foundational probability rules such as addition and multiplication.
• Develop Computational Proficiency for Statistical Analysis: 
  Apply Python programming to visualize data, compute statistical measures, simulate experiments, and analyze both discrete and continuous probability distributions effectively.
• Analyze Random Variables and Distributions: 
  Explore discrete and continuous random variables, compute key properties like expectation and variance, and utilize probability distributions to model real-world scenarios.
• Apply Theoretical Concepts to Real-World Problems: 
  Use sampling techniques, explore the Central Limit Theorem, and implement hypothesis testing to make data-driven inferences and validate statistical claims.
• Integrate Theoretical and Practical Approaches in Statistical Analysis: 
  Combine theoretical knowledge with hands-on Python programming to analyze joint and marginal distributions, visualize complex statistical relationships, and solve advanced problems in probability and statistics.

After completing this course, students should be able to:

• Descriptive and Inferential Statistics: 
  Develop a strong understanding of statistical foundations, including data visualization, measures of location, variability, and the application of descriptive statistical methods using Python.
• Probability Theory and Applications: 
  Gain proficiency in basic and advanced probability concepts, including conditional probability, Bayes' Theorem, and the implementation of probability rules and distributions through Python programming.
• Discrete and Continuous Distributions: 
  Analyze and simulate discrete and continuous random variables, compute expectations and variances, and understand the application of distributions such as Binomial, Poisson, Normal, and more in real-world scenarios.
• Sampling and Statistical Inference: 
  Explore the principles of random sampling, sampling distributions, and the Central Limit Theorem, enabling statistical reasoning for analyzing sample means and variances in various contexts.
• Hypothesis Testing and Decision-Making: 
  Master hypothesis testing concepts, including the null hypothesis, p-values, type I and II errors, and apply statistical tests to real-world problems, interpreting results with confidence and accuracy.

• Devore, JL, Probability & Statistics for Engineering and the Sciences, 8th Edition, Cengage Learning, 2012.

• Milton, JS and Arnold JC, Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences, 4th Edition, Tata McGraw-Hill, 2007.
• Walpole, RE, Myers, RH, Myers, SL, Ye, KE, Probability & Statistics for Engineers and Scientists, 9th Edition, Pearson Education, 2016.
• Johnson, RA, Miller Freund’s Probability and Statistics for Engineers, 8th Edition, PHI, 2010.
• Meyer, PL, Introductory Probability and Statistical Applications, 2nd Edition, Addison-Wesley, 1970.
• Ross, SM, Introduction to Probability Models, 11th Edition, Academic Press, 2014.