Mathematical Prerequisites¶
This appendix is a self-contained reference for the mathematics, probability, and statistics that the course builds on. It is not a detour — quantitative trading is applied probability and statistics, and Part III — Statistics for Trading links directly into these pages wherever a lesson leans on a formal result.
How to use this appendix¶
- If you have a quantitative background, skim the part titles and move on. Come back when a course lesson cites a specific result.
- If you are newer to probability, work through Parts I–VIII in order — they build from counting principles up to the Central Limit Theorem and Markov processes, which is exactly the toolkit regime models and Monte Carlo methods assume later.
- The statistics parts (X–XVII) cover estimation, hypothesis testing, regression, model selection, multiple testing, Bayesian methods, and the numerical machinery that makes all of it computable — the machinery behind every "is this strategy actually profitable?" question the course asks.
- Part XVIII connects the formal results back to trading: sizing, risk, microstructure, and regime detection. It is where the appendix argues for itself, since every page there finds a model whose own diagnostic cannot see the thing it got wrong.
Contents¶
| Part | Focus |
|---|---|
| Part I — Mathematical Foundations | Sets, counting, notation, series, linear algebra, calculus, and compound growth |
| Part II — Foundations of Probability | Sample spaces, axioms, conditioning, Bayes' rule, independence, and the law of total probability |
| Part III — Random Variables | CDFs, PMFs, PDFs, joint and conditional distributions, transformations |
| Part IV — Expectation and Moments | Expectation, variance, skewness and kurtosis, covariance and correlation, conditional expectation, and the total laws |
| Part V — Common Probability Distributions | The standard discrete and continuous families |
| Part VI — Multivariate Probability | Random vectors, covariance matrices, and the multivariate Gaussian |
| Part VII — Asymptotic Theory | Laws of large numbers, the CLT, and the mapping theorems |
| Part VIII — Stochastic Processes | Arrival processes, Markov chains, HMMs, and Brownian motion |
| Part IX — Monte Carlo Methods | Simulation, sampling, variance reduction, and resampling |
| Part X — Foundations of Statistics | Samples, sampling distributions, and statistical models |
| Part XI — Parameter Estimation | Point and interval estimation, ML, moments, and Bayesian estimates |
| Part XII — Hypothesis Testing | Tests, p-values, errors, power, and the main test families |
| Part XIII — Regression and Statistical Models | Linear and generalized linear models, regularization, diagnostics |
| Part XIV — Model Selection | Bias–variance, cross-validation, information criteria |
| Part XV — Multiple Testing | Corrections for many comparisons and data snooping |
| Part XVI — Bayesian Statistics | Priors, posteriors, conjugacy, updating, model comparison, and prediction |
| Part XVII — Statistical Computing | Optimization, EM, and MCMC |
| Part XVIII — Applications to Quantitative Finance | Kelly and ruin, drawdowns and passage times, queues and order flow, evidence, simulated prices and risk, VaR and expected shortfall, tails, copulas, and regime detection |
Note
Much of this material was migrated from an earlier site and is kept mathematically rigorous, with proofs in collapsible blocks. Every part carries seeded Python whose output is pinned in the source, so every number on the page can be reproduced by running the block above it. No page reads cached market data, so the appendix runs anywhere numpy and scipy do.