what happens as you sample more
Laws.
Patterns that show up once you stop trusting a single draw.
13 live pieces
· 15
more on the way
Central Limit Theorem
Average almost anything enough times and it starts looking Gaussian.
Law of Large Numbers
Running averages settle. Watch 20 coin-flip runs converge to the same value.
Bayes' theorem
Update beliefs when evidence arrives. Area diagram makes posterior probability visible.
Random walk
Each step is ±1. Spread grows as √t — not t. See what bias does to the drift.
Regression to the mean
Extreme values pull back toward average — not because of any force, but pure probability.
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.
Delta method
soon Linearize a nonlinear function of an estimator. Watch the approximation break.
Law of total variance
soon Var(Y) splits into within-group and between-group. A decomposition you can bar-chart.
Tower property
soon E[E[Y | X]] = E[Y]. Iterated expectation — conditioning averages up cleanly.
Maximum likelihood
soon The log-likelihood surface with a draggable parameter. Gradients point to the MLE.
Fisher information
soon Curvature of the log-likelihood. Sharper peak, smaller variance — that’s Cramér-Rao.
Shannon entropy
soon The bits needed to describe a random variable. Maximized by uniform, minimized by certainty.
KL divergence
soon How far apart are two distributions? Directional, never symmetric, always ≥ 0.
Cross-entropy
soon The loss function behind classification. Breaks into entropy plus KL.
Poisson process
soon Arrivals on a timeline. Gaps are exponential; counts in windows are Poisson.
Brownian motion
soon Scale a random walk and it goes continuous. The limit is Wiener.
Martingales
soon Fair games that stay fair under any stopping rule. Optional stopping, made visual.
Conjugate priors
soon Beta-Binomial and Gamma-Poisson updating live. Watch the posterior sharpen.
Kelly criterion
soon Bet-sizing on a biased coin. Under, over, and exactly optimal wealth trajectories.
Branching process
soon Each member has a random number of offspring. Extinction vs. explosion.
Glivenko-Cantelli
soon The ECDF converges uniformly to the true CDF. The fundamental theorem of statistics.