# 30-Day Probability & Statistics > Daily (5d/wk): 45m concepts (canonical externals) + 45m problems + 15m notes. Every week ships code or a writeup. External-heavy (Seeing Theory, MIT OCW, StatQuest, ISL). Not aesthetic math — the gym is solving, simulating, checking assumptions. Canonical: https://learn.significanthobbies.com/curriculum/roadmaps/prob-stats-30d Daily (5d/wk): 45m concepts (canonical externals) + 45m problems + 15m notes. Every week ships code or a writeup. External-heavy (Seeing Theory, MIT OCW, StatQuest, ISL). Not aesthetic math — the gym is solving, simulating, checking assumptions. - Horizon: 30d - Outcome: Active probability & statistics — solve, derive, implement. No passive video watching as a substitute for problems. - Tracks: Mathematics Work each milestone as a claim that needs evidence. Predict the mechanism and failure mode first, study the linked concepts, complete the drills without copying an answer, and preserve the resulting code, measurements, or design. ## Milestone 1: Prerequisites — unlock the rest of this path Concepts the later milestones depend on. Without these the planner cannot serve the rest of this roadmap. ### Concepts - [Combinatorics](https://learn.significanthobbies.com/curriculum/concepts/combinatorics) — Counting without enumerating — permutations, combinations, inclusion-exclusion, pigeonhole. - [Vectors & Vector Spaces](https://learn.significanthobbies.com/curriculum/concepts/vectors-and-spaces) — Vectors as ordered lists, dot products, norms, orthogonality, and the geometric picture of n-dimensional space. - [Derivatives & Gradients](https://learn.significanthobbies.com/curriculum/concepts/derivatives-and-gradients) — Partial derivatives, the gradient vector, and reading a loss surface for descent direction. - [Matrices & Linear Transformations](https://learn.significanthobbies.com/curriculum/concepts/matrices-and-transformations) — Matrix multiplication as composing linear maps: rotation, scaling, projection, and change of basis. ## Milestone 2: Week 1 — Probability Conditional probability, distributions, expectation — the language of uncertainty. ### Concepts - [Probability Fundamentals](https://learn.significanthobbies.com/curriculum/concepts/probability-fundamentals) — Sample spaces, conditional probability, independence, and Bayes' rule for updating beliefs. - [Random Variables & Distributions](https://learn.significanthobbies.com/curriculum/concepts/random-variables) — Discrete and continuous distributions, expectation, variance, and the law of large numbers. - [Classical Distributions](https://learn.significanthobbies.com/curriculum/concepts/classical-distributions) — Bernoulli, Binomial, Poisson, Normal, and Exponential — when each models the world and what to expect from them. ### Build evidence - **Simulate coin flips, dice, and random walks** — Code simulators (no stats libraries): Bernoulli trials, dice sums, and a 1D random walk. Plot or print distributions and compare sample mean to theory. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 3: Week 2 — Summaries & sampling Describe data honestly; understand why averages become Normal. ### Concepts - [Descriptive Statistics](https://learn.significanthobbies.com/curriculum/concepts/descriptive-statistics) — Summarizing data: mean, median, variance, correlation, and when each summary lies. - [Covariance & Correlation](https://learn.significanthobbies.com/curriculum/concepts/covariance-correlation) — Covariance measures co-movement; correlation normalizes to [−1, 1]. Foundation for regression, PCA, and portfolio risk. - [Sampling & the Central Limit Theorem](https://learn.significanthobbies.com/curriculum/concepts/sampling-and-clt) — Sampling distributions, standard error of the mean, and why averages become Normal as n grows. - [Estimation & Confidence Intervals](https://learn.significanthobbies.com/curriculum/concepts/estimation-confidence) — Point estimates, standard error, confidence intervals, and what '95% confident' actually means. ### Build evidence - **Implement mean, variance, correlation from scratch** — On a real CSV (or synthetic data), compute mean, variance, std dev, covariance, and Pearson r without pandas/numpy — loops only. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 4: Week 3 — Inference & experiments Hypothesis tests, regression, and real A/B practice for SWE. ### Concepts - [Hypothesis Testing](https://learn.significanthobbies.com/curriculum/concepts/hypothesis-testing) — Null and alternative hypotheses, p-values, significance, power, and Type I/II errors. - [A/B Testing for Engineers](https://learn.significanthobbies.com/curriculum/concepts/ab-testing-engineering) — Sample size, statistical power, practical significance, SRM checks, and multiple-comparison traps in product experiments. - [Linear Regression](https://learn.significanthobbies.com/curriculum/concepts/regression-basics) — Fitting a line (or hyperplane) by least squares; residuals, R², and the geometry of projection. - [Bias, Variance & Overfitting](https://learn.significanthobbies.com/curriculum/concepts/bias-variance-overfitting) — Underfitting vs memorizing noise; why in-sample greatness lies. ### Build evidence - **A/B test analysis report** — Analyze a mock or real experiment: check SRM, compute CI on the lift, interpret p-value and practical significance, and write a ship/no-ship recommendation. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 5: Week 4 — Bayesian & likelihood Posterior thinking and MLE — the bridge to ML loss functions. ### Concepts - [Bayesian Inference](https://learn.significanthobbies.com/curriculum/concepts/bayesian-inference) — Priors, posteriors, credible intervals, and when Bayesian updating beats frequentist tests. - [Maximum Likelihood Estimation](https://learn.significanthobbies.com/curriculum/concepts/maximum-likelihood) — Choosing parameters that make the observed data most probable; log-likelihood; connection to cross-entropy loss. - [Information & Entropy](https://learn.significanthobbies.com/curriculum/concepts/information-entropy) — Entropy as surprise, cross-entropy as a loss, KL divergence as a distributional distance. ### Build evidence - **Analyze a fake trading signal** — Given (or generate) a spurious 'alpha' signal on synthetic or historical noise, show why it fails: multiple testing, in-sample overfit, or non-stationarity. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making.