Part I — Mathematical Foundations¶
The mathematical language the rest of the appendix is written in. Nothing here is probabilistic: these are the tools the probability starts from — sets and functions, counting, notation, limits, linear algebra, calculus, and the exponential-versus-logarithm arithmetic that all compound growth reduces to.
Read selectively. If a course lesson sends you here for one result, take that result and leave; the pages are written to be entered in the middle. If you are building up to Part II and beyond from scratch, the dependencies run roughly in file order, with two exceptions worth knowing: Sequences and Infinite Series supplies the limits that Calculus Essentials assumes, and Basic Linear Algebra Review supplies the matrix gradients that the same page's optimization section uses.
Topics¶
| Topic | Focus |
|---|---|
| Sets and Functions | Set operations, de Morgan's laws, partitions, preimages, and indicators — why an event is a set and a random variable is a function |
| Counting Principles | Multiplication principle, permutations, combinations, inclusion–exclusion, and the counting explosion behind backtest overfitting |
| Mathematical Notation | The symbol contract for the whole appendix: sums and products, probability and estimator notation, matrix conventions, asymptotic notation |
| Sequences and Infinite Series | Convergence, limit laws, series tests, and the geometric series behind exponential weighting |
| Basic Linear Algebra Review | Vectors, matrices, quadratic forms, eigenvalues, conditioning, Cholesky, and least squares as a projection |
| Calculus Essentials | Derivatives, Taylor expansion, integration, the Gaussian integral, and constrained optimization with Lagrange multipliers |
| Exponentials, Logarithms, and Growth | Compounding, log versus simple returns, volatility drag, \(\sqrt{T}\) scaling, half-lives, and why prices are lognormal |