Probability
Aug 30, 2024
Probability and Statistics
28 min readProbability and Statistics (8): Bayesian Statistics — Priors, Posteriors, and Why Frequentists Argue
Bayesian inference from first principles: posterior distributions, conjugate priors, the Beta-Binomial and Normal-Normal models, credible intervals, predictive distributions, MCMC intuition, and deep connections to …
Aug 28, 2024
Probability and Statistics
28 min readProbability and Statistics (7): Hypothesis Testing — p-Values, Confidence Intervals, and All Their Pitfalls
A rigorous treatment of hypothesis testing, p-values, Type I/II errors, confidence intervals, and multiple testing corrections — including the misinterpretations that trip up even experienced practitioners, with Python …
Aug 26, 2024
Probability and Statistics
26 min readProbability and Statistics (6): Estimation — MLE, MAP, and the Bias-Variance Story
Point estimation from method of moments through maximum likelihood and MAP, with Fisher information, the Cramer-Rao bound, and the bias-variance decomposition that explains overfitting and underfitting.
Aug 24, 2024
Probability and Statistics
28 min readProbability and Statistics (5): Law of Large Numbers and the Central Limit Theorem
The two pillars of probability: the Law of Large Numbers guarantees sample means converge, and the Central Limit Theorem explains why everything looks Gaussian — with proofs, convergence concepts, and Python simulations.
Aug 23, 2024
Probability and Statistics
32 min readProbability and Statistics (4): Joint Distributions, Marginalization, and Independence
Joint PMFs and PDFs, marginal and conditional distributions, the bivariate normal, transformations via the Jacobian method, convolutions, and order statistics — with proofs and contour plot visualizations.
Aug 21, 2024
Probability and Statistics
26 min readProbability and Statistics (3): Expectation, Variance, and the Moment-Generating Trick
From expectation and variance through covariance, correlation, and moment-generating functions to Chebyshev's inequality — the complete toolkit for summarizing random variables, with proofs for every result.
Aug 20, 2024
Probability and Statistics
28 min readProbability and Statistics (2): Random Variables and the Distributions That Matter
A rigorous tour of random variables, PMFs, PDFs, CDFs, and every distribution that matters in practice — Bernoulli, Binomial, Poisson, Gaussian, Exponential, Gamma, and Beta — with derivations, proofs, and Python …
Aug 18, 2024
Probability and Statistics
36 min readProbability and Statistics (1): Probability Spaces — Why We Need Axioms (But Won't Overdo It)
Building probability from the ground up: sample spaces, Kolmogorov's axioms, conditional probability, Bayes' theorem, and the birthday problem — with proofs and Python simulations.