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Statistics

Probability distributions, random variables, Central Limit Theorem, hypothesis testing, confidence intervals, and Bayesian inference.

1Probability Axioms, Random Variables & Expected Value Mechanics

Kolmogorov probability axioms, joint/conditional probabilities, Bayes' Theorem, and variance derivations.

Probability Foundations & Bayes' Theorem Architecture

Axiomatic probability spaces, conditional independence, and Bayesian posterior updating.

Kolmogorov Axioms, Conditional Probability & Bayes' Theorem Posterior Updating

Kolmogorov's three probability axioms, conditional probability definitions, Law of Total Probability, and Bayes' Theorem updating.

2Continuous Distributions, Moment Generating Functions & the CLT

Continuous random variables, probability densities, moment generating functions, Law of Large Numbers, and Central Limit Theorem convergence.

Probability Density Functions, MGFs & Central Limit Dynamics

Probability density integration, moments of distributions, MGF uniqueness, and asymptotic normality proofs under the Central Limit Theorem.

Continuous Densities, Moment Generating Functions & the Central Limit Theorem

Master probability density functions, cumulative distribution geometry, Moment Generating Functions (MGFs), expectation/variance derivations via derivatives, and the analytical proof of the Central Limit Theorem.

3Parametric Point Estimation, Maximum Likelihood & Fisher Information

Point estimation, bias and consistency, score functions, Maximum Likelihood Estimation (MLE), Fisher Information, and the Cramér-Rao Lower Bound.

Likelihood Functions, Score Dynamics & Efficiency

Log-likelihood maximization, Newton-Raphson scoring, Fisher Information matrix, and minimum-variance unbiased estimators.

Maximum Likelihood Estimation, Score Functions & the Cramér-Rao Lower Bound

Master likelihood construction, log-likelihood optimization, score equations, the Fisher Information Matrix, the Cramér-Rao Lower Bound, and asymptotic normality of MLE estimators.

4Hypothesis Testing, Neyman-Pearson & Bayesian MCMC Inference

Neyman-Pearson Lemma, statistical power, p-values, Benjamini-Hochberg FDR control, and Markov Chain Monte Carlo (MCMC) sampling.

Hypothesis Testing, FDR Control & Metropolis-Hastings MCMC

Likelihood ratio tests, Neyman-Pearson lemma, multiple comparison adjustments (FDR), and Bayesian posterior sampling via Metropolis-Hastings.

Neyman-Pearson Likelihood Ratio Tests, Benjamini-Hochberg FDR & Metropolis-Hastings MCMC

Master statistical hypothesis testing rigor, Type I and II error tradeoffs, the Neyman-Pearson Lemma for most powerful tests, false discovery rate (FDR) control, and Bayesian Markov Chain Monte Carlo (MCMC) sampling.