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Foundations of Probability

  • What is Probability?
  • Theoretical vs Empirical Probability
  • Three Views of Probability
  • Sample Space and Events
  • Axioms of Probability
  • Independence and Expectation
  • Variance and Standard Deviation
  • Covariance and Correlation
  • Key Inequalities

Set Theory & Combinatorics

  • Set Operations in Probability
  • Counting Methods
  • Advanced Counting

Conditional & Bayesian Probability

  • Conditional Probability
  • Bayes' Theorem
  • Law of Total Probability

Random Variables & Distributions

  • What is a Random Variable?
  • Discrete vs Continuous
  • PDFs and CDFs
  • Expectation, Variance, and Moments

Discrete Distributions

  • Bernoulli and Binomial
  • Poisson and Geometric
  • Negative Binomial and Hypergeometric

Continuous Distributions

  • Uniform and Normal
  • Exponential, Gamma, Beta
  • Heavy-Tailed Distributions

Limit Theorems

  • Law of Large Numbers
  • Central Limit Theorem
  • Convergence in Probability vs Distribution

Frequentist Inference

  • Confidence Intervals
  • Hypothesis Testing
  • p-values and Statistical Decisions
  • Type I and Type II Errors
  • Power and Effect Size
  • Bootstrapping and Resampling

Advanced Probability Tools

  • Law of the Unconscious Statistician
  • Moment Generating Functions
  • Characteristic Functions
  • Markov Chains
  • Stationary Distributions

Bayesian Inference

  • Bayesian Philosophy
  • Prior, Likelihood, Posterior
  • Conjugate Priors
  • MCMC and Modern Computation

Regression Analysis

  • Ordinary Least Squares
  • Multiple Linear Regression
  • Regression Diagnostics
  • Regularization
  • Logistic and Generalized Linear Models

Multivariate Statistics

  • Joint, Marginal, and Conditional
  • Multivariate Normal
  • Covariance Matrices
  • Correlation vs Causation
  • Principal Component Analysis

Stochastic Processes

  • Random Walks
  • Poisson Processes
  • Brownian Motion
  • Itô's Lemma
  • Martingales
  • Geometric Brownian Motion

Simulation & Approximation

  • Monte Carlo Simulation
  • Variance Reduction
  • Bootstrapping for Finance
  • Quasi-Monte Carlo

Time Series

  • Stationarity and Autocorrelation
  • AR, MA, and ARIMA
  • GARCH and Volatility Clustering
  • Cointegration and Pairs Trading
  • Kalman Filters

Information Theory

  • Shannon Entropy
  • Kullback–Leibler Divergence
  • Mutual Information
  • Maximum Entropy

Linear Algebra

  • Vectors, Norms, and Inner Products
  • Matrix Operations
  • Eigenvalues and Eigenvectors
  • Singular Value Decomposition
  • Positive Definite Matrices
  • Numerical Stability

Calculus & Optimization

  • Multivariate Calculus
  • Lagrange Multipliers
  • Convex Optimization
  • Gradient Descent and Variants
  • Stochastic Calculus Primer

Machine Learning Fundamentals

  • Supervised vs Unsupervised
  • Bias–Variance Trade-off
  • Cross-Validation
  • Tree-Based Methods
  • Support Vector Machines
  • Clustering and Dimensionality Reduction
  • Classification Metrics

Deep Learning

  • Feedforward Networks
  • Backpropagation
  • Optimizers and Schedules
  • Regularization in DL
  • Architectures for Finance
  • Loss Functions

Options Pricing

  • Payoffs and Put–Call Parity
  • Risk-Neutral Valuation
  • Binomial Trees
  • Black–Scholes
  • The Greeks
  • Volatility Smile and Surface
  • Exotic Options

Portfolio Theory

  • Mean–Variance Optimization
  • CAPM and Factor Models
  • Sharpe, Sortino, and Information Ratio
  • Black–Litterman
  • Risk Parity

Trading & Risk Applications

  • Value-at-Risk
  • Expected Shortfall
  • Backtesting
  • Market Making Basics
  • Execution and Market Microstructure
  • Statistical Arbitrage
Study Guide/Deep Learning
Section 20 · Lesson 20.94

Regularization in DL

Dropout, weight decay, and early stopping.

Deep nets have so many parameters that regularization isn't optional.

Weight decay penalizes ∥θ∥2\|\theta\|^2∥θ∥2, equivalent to L2L_2L2​ regularization. It's almost always worth using; modern optimizers like AdamW separate it from the gradient update for cleaner behavior.

Dropout randomly zeros out a fraction of activations during training, forcing the network not to rely on any single neuron. It's roughly equivalent to ensembling many subnetworks. Use lower dropout in convolutional layers, higher (e.g. 0.50.50.5) in fully-connected ones.

Early stopping monitors validation loss and stops training when it stops improving — a free regularizer that also saves compute.

Data augmentation, batch normalization, and label smoothing are also common regularizers in production deep-learning pipelines.

Why does dropout help generalization?

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Optimizers and Schedules
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Architectures for Finance