Probability describes the uncertainty
Begin with random variables, distributions, conditional expectation, variance and covariance. In finance, the conditioning information matters: a forecast at time t must use information available by time t. Distinguish a population quantity from an estimate based on a finite, dependent sample.
A sample average return is an estimate, not a fixed property of an asset. Its uncertainty depends on sample size, serial dependence, the distribution of outcomes and changes in the environment.
Linear algebra connects assets and features
Vectors represent exposures or observations; matrices represent features, transformations and covariance. Understand rank, conditioning, projections, eigenvalues and positive semidefinite matrices before relying on regression or covariance optimisation.
Portfolio variance includes covariance terms. Adding more positions does not necessarily diversify the risks that matter. Poorly estimated covariance matrices can also create unstable optimised weights.
Statistics makes the comparison meaningful
Study estimation, confidence intervals, hypothesis testing, regression diagnostics and model selection. Ask whether the null hypothesis answers the actual research question. Failure to reject a null is not proof that it is true, and a small p-value does not quantify a trading opportunity.
Time series adds order
Returns, volatility, volume and spreads can have different persistence and seasonal behaviour. Learn autocorrelation, conditional heteroskedasticity, stationarity assumptions, structural breaks and state-space models. Fit transformations inside the training window, then apply them forward.
A first practical exercise
Using a documented dataset, calculate simple and log returns, a trailing mean, a trailing standard deviation and drawdowns. Repeat the estimates across chronological blocks. Explain which differences could reflect sampling uncertainty and which require a change in the model.
Continue with stochastic calculus for continuous-time models, or use the reading collection to explore the source references.