what happens as you sample more
Laws.
Patterns that show up once you stop trusting a single draw.
22 live pieces
Central Limit Theorem
See when standardized averages approach a normal shape, and when they do not.
Law of Large Numbers
Compare coin and Cauchy averages, exact error probabilities, and nonmonotone sample paths.
Bayes' theorem
Update beliefs when evidence arrives. Area diagram makes posterior probability visible.
Random walk
Separate drift, spread, and distance; inspect exact endpoint probabilities and seeded paths.
Regression to the mean
Select extreme first readings and compare repeated measurements under an explicit noise model.
Markov chains
Wander between states with fixed transition probabilities. Long-run averages stabilize.
Markov's inequality
P(X ≥ a) ≤ E[X]/a. The simplest tail bound, often the loosest.
Chebyshev's inequality
Tail probability bounded by 1/k². Drag k and watch how loose it really is.
Jensen's inequality
For convex f: E[f(X)] ≥ f(E[X]). Watch the gap grow as you bend the curve.
Hoeffding's inequality
Concentration of bounded random variables. The bound tightens as n grows.
The bootstrap
Resample from your sample. Standard errors without any formula.
Confidence intervals
100 experiments, roughly 95 ribbons cover μ. Fixes the usual misreading.
Hypothesis testing
Null distribution with a sliding observed statistic. P-values as tail areas.
Law of total variance
Separate within-group and between-group spread, then change the grouping.
Maximum likelihood
Hold observations fixed and inspect interior, boundary, and nonexistent likelihood maxima.
Shannon entropy
Inspect expected surprise, code lengths, and the uncertainty of a finite source.
KL divergence
Compare directional expected log-loss, signed contributions, and support failures.
Cross-entropy
Separate source uncertainty from prediction error using expected log-loss.
Poisson process
Arrivals on a timeline. Gaps are exponential; counts in windows are Poisson.
Brownian motion
Refine a consistent Gaussian time grid and inspect diffusion and quadratic variation.
Martingales
Test conditional fairness, bounded stopping, and the contribution of unfinished paths.
Conjugate priors
Update beta and gamma priors, then integrate uncertainty into predictions.
Need a starting point? Follow a learning path.
Planned topics (6)
These topics are on the editorial backlog. Publication dates are not set.
- Delta method
- Tower property
- Fisher information
- Kelly criterion
- Branching process
- Glivenko-Cantelli