Probability and Statistics: The Science of Uncertainty

A comprehensive introductory university-level course on the mathematical foundations of probability and statistics. Requiring one year of calculus, the course covers probability models, random variables, expectation, sampling distributions, likelihood and Bayesian inference, and relationships among variables.

Master the rigorous mathematical science of uncertainty through calculus-based probability and statistical inference.

Author: Michael J. Evans and Jeffrey S. Rosenthal

Acknowledgments: The authors acknowledge contributions from various reviewers and colleagues at institutions such as the University of Toronto, McMaster University, and Purdue University. Funding and infrastructure support from the University of Toronto are also noted.

1. Define a formal probability model using sample spaces, events, and probability measures.
2. Apply combinatorial principles (permutations, subsets, binomial coefficients) to solve uniform probability problems.
3. Utilize the Law of Total Probability and Bayes' Theorem to analyze multi-stage systems and update beliefs based on new information.
4. Define and distinguish between discrete and absolutely continuous random variables and their respective probability/density functions.
5. Identify and apply key probability distributions (Bernoulli, Binomial, Poisson, Normal, etc.) to model real-world phenomena.
6. Compute marginal densities, conditional distributions, and assess independence for multivariate distributions.
7. Calculate the expected value, variance, and covariance for discrete, continuous, and mixed random variables.
8. Apply the Law of the Unconscious Statistician (LOTUS) and linearity properties to compute expectations of transformed variables.
9. Derive moments using Probability-Generating (PGF) and Moment-Generating Functions (MGF).
10. Define and derive sampling distributions for functions of i.i.d. sequences.