Phase 7
Probability Theory
Phase 7 of the Quant Academy curriculum.
7.1Full Lesson
Probability Spaces, Random Variables, and Distributions
The measure-theoretic scaffolding: sample spaces, sigma-algebras, probability measures, measurable maps, and the laws th
7.2Full Lesson
Expectation, Moments, and Key Inequalities
Expectation as the Lebesgue integral of a random variable, its moments, and the Markov, Chebyshev, Jensen, and Cauchy-Sc
7.3Full Lesson
Independence, Conditional Probability, and Conditional Expectation
From elementary conditioning and Bayes to conditional expectation as a projection onto a sigma-algebra - the objec
7.4Full Lesson
Common Distributions and Transform Methods (MGF, Characteristic Functions)
The standard catalogue of laws and the transforms - moment generating and characteristic functions - that ma
7.5Full Lesson
The Laws of Large Numbers
Why sample averages converge to expectations - the weak law via Chebyshev, the strong law almost surely, and the h
7.6Full Lesson
The Central Limit Theorem and Modes of Convergence
Why standardized sums are universally Gaussian, proved through characteristic functions, and the hierarchy of convergenc
7.7Full Lesson
Martingales and Stopping Times (Discrete Time)
Filtrations, the fair-game condition, stopping times, and the optional stopping theorem - the discrete-time skelet