Mathematical Foundations

The measure-theoretic triple $(\Omega, \mathcal{F}, \mathbb{P})$ that makes probability rigorous, why we need σ-algebras, what Kolmogorov's axioms buy us, and where naive "probability = favourable/total" quietly breaks.

Random variables as measurable functions that push a probability measure forward into a distribution, CDFs, densities, the change-of-variables formula for transformations, and the joint/marginal/conditional structure that underlies every multivariate model.

Expectation as the Lebesgue integral against a distribution, LOTUS, the algebra of variance and covariance, and the higher moments (skewness, kurtosis) that decide whether a Gaussian risk model is safe or lethal.

The core distribution family every quant carries in their head, Bernoulli through Binomial, Poisson, Normal, Exponential and Lognormal, with the limiting relationships that connect them and the moment structure that decides where each one fits.

The transform $M_X(t) = \mathbb{E}[e^{tX}]$ that encodes every moment in one function, with its uniqueness theorem, the sums-of-independents multiplication rule, and cumulants, the additive cousins that linearise skewness and kurtosis.

Why sample averages converge to the mean, the weak vs. strong versions and their modes of convergence, proof sketches via Chebyshev, and the practical ceiling it puts on Monte Carlo accuracy and any edge-based trading.

Why sums of independent shocks become Gaussian, the classical statement and its MGF proof, the Lindeberg/Lyapunov conditions that decide when it applies, the Berry–Esseen bound on its speed, and the fat-tailed reality that makes finance its most dangerous misuse.

The rule for inverting conditional probability, prior to posterior, the base-rate trap that fools intuition, conjugate updating that makes it tractable, and the sequential/log-odds forms that turn belief revision into arithmetic.

Memoryless stochastic processes governed by a transition matrix, the stationary distribution as a left eigenvector, the ergodic theorem that makes long-run averages predictable, and hitting-time systems that price first-passage problems.

The formal model of a fair game, conditional expectation, the optional stopping theorem and why "no betting system beats a fair game", and the martingale view of arbitrage-free pricing.

The vector-space machinery that every quantitative model runs on, subspaces and rank, orthogonal projection as the geometry of regression, matrix and vector norms, and the quadratic forms that encode portfolio risk.

The invariant directions of a linear map, the spectral theorem for symmetric matrices, diagonalization as a change to the natural basis, and power iteration, the algorithm that turns eigen-structure into PCA and PageRank.

The universal factorization $A = U\Sigma V^\top$ that works for any matrix, its geometry as rotate-stretch-rotate, the Eckart–Young optimal low-rank approximation behind PCA and denoising, and the pseudo-inverse that solves least squares even when nothing is invertible.

The matrices that make quadratic forms non-negative, equivalent tests via eigenvalues, pivots and principal minors, the Cholesky factorization that generates correlated risk, and why every valid covariance matrix must be PSD.

The class of problems where every local optimum is global, convex sets and functions, the first-order optimality condition, and Lagrangian duality with the weak/strong duality gap that underlies robust portfolio construction.

The calculus of constrained optimization, Lagrange multipliers for equalities, the full Karush–Kuhn–Tucker conditions with complementary slackness for inequalities, and their application to the mean–variance efficient frontier.