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Interactive Explainers

Play with the theorems

Interactive explainers that let you move the mathematics: drag ε and watch δ answer, walk the topologist's sine curve, trap a subsequence by bisection, watch Newton's method fall into a cycle. 15 are free to play right here — the rest come with Trap Analysis.

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Video explainers

Narrated walk-throughs of the trickiest ideas, made for this site — no ads, no recommendations pulling you elsewhere. Free to watch whether or not the interactive below is unlocked.

Unit 4Limit Theorems and Markov Chains › Modes of convergence, WLLN, SLLN, CLT#

The Central Limit Theorem, watchedinteractivefree

Unit 4Hypothesis Testing › Neyman–Pearson lemma and UMP tests#

Type I against Type II: the trade-off you cannot escapeinteractivefree

Unit 4Probability › Axioms, conditional probability, independence, Bayes#

Bayes and the base rate: why a 99% accurate test is usually wronginteractivefree

Step 1 / 4The setup

A test has sensitivity 0.990.99 and specificity 0.950.95; the disease affects 1%1\% of people. You test positive. What is the chance you have it?

Unit 4Probability › Standard discrete and continuous distributions#

A distribution's own shape, as its parameters moveinteractivefree

Unit 4Limit Theorems and Markov Chains › Markov chains: classification of states, stationary distributions#

Why periodic chains never settleinteractive

This visual is part of Trap Analysis

A cycle with stationary distribution (⅓,⅓,⅓) that never converges to it — until one slider makes it aperiodic.

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Unit 4Hypothesis Testing › Likelihood ratio and standard tests#

What 95% confidence actually meansinteractive

This visual is part of Trap Analysis

Forty samples, forty intervals, and a count of how many cover the truth — the misconception this kills is worth several marks.

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Unit 4Probability › Random variables, distributions, moments, MGF#

Moments, MGFs, and when they fail to existinteractive

This visual is part of Trap Analysis

The tail-integral formula, the two inequalities worth memorising, and a distribution with no mean at all.

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Unit 4Probability › Standard discrete and continuous distributions#

The signatures that identify a distribution instantlyinteractive

This visual is part of Trap Analysis

One characterising property each for Poisson, exponential and normal.

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Unit 4Probability › Joint distributions, transformations, order statistics#

Order statistics from first principlesinteractive

This visual is part of Trap Analysis

Derive the max, the min and the k-th — with the dependence that catches people out.

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Unit 4Linear Models and Multivariate › Multivariate normal distribution#

Zero correlation and independence: when they coincideinteractive

This visual is part of Trap Analysis

The equivalence holds under joint normality and fails without it — with the standard counterexample.

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Unit 4Estimation › MLE and method of moments#

When the MLE is not found by differentiatinginteractive

This visual is part of Trap Analysis

The uniform case, where calculus finds nothing and the answer sits on the boundary.

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Unit 4Estimation › Sufficiency, completeness, UMVUE, Cramér–Rao#

The Lehmann–Scheffé pipelineinteractive

This visual is part of Trap Analysis

Sufficient, then complete, then unbiased — the three steps that produce a UMVUE.

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Unit 4Linear Models and Multivariate › Gauss–Markov, regression, ANOVA basics#

What Gauss–Markov assumes, and what it does notinteractive

This visual is part of Trap Analysis

BLUE without any normality — and what normality actually buys you.

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Unit 4Sampling and Design of Experiments › SRS, stratified and systematic sampling#

Why stratification helps and systematic sampling can betray youinteractive

This visual is part of Trap Analysis

The variance decomposition behind stratification, and the periodicity that ruins the systematic scheme.

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Unit 4Sampling and Design of Experiments › CRD, RBD, LSD essentials#

CRD, RBD and Latin square: degrees of freedom without memorisinginteractive

This visual is part of Trap Analysis

Derive each error df by subtraction, and know when blocking costs more than it buys.

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Every interactive explainer comes with Trap Analysis — along with the worked solutions, the per-option trap diagnosis and the counterexample objects. The papers, the book and the video map are free without it.

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