Statistics

Extend single-variable calculus into 3D with multivariable functions, vectors, and surface integrals.

Work with matrices, vectors, and systems of equations used in graphics, AI, and engineering.

Prove the mathematical laws governing randomness, from coin flips to continuous distributions.

Derive estimators, test statistics, and confidence intervals from probability theory foundations.

Write code in R or Python to implement statistical methods, simulations, and data analysis.

Model relationships between variables and predict outcomes using linear and logistic regression.

Plan experiments with proper controls, randomization, and sample sizes to get trustworthy results.

Analyze datasets with many variables simultaneously using PCA, clustering, and factor analysis.

Rigorously prove foundational theorems about limits, sequences, and continuity underlying calculus.

Analyze data collected over time — trends, seasonality, and forecasting in financial or scientific data.

Update beliefs with new evidence using Bayes' theorem and prior/posterior probability distributions.

Design survey samples and estimate population parameters with proper margin of error.

Test hypotheses without assuming data follows a specific distribution — rank tests and bootstrap methods.