A statistics learning path that starts from NTU IM exam preparation, builds through statistical inference and applied modeling, and connects each topic to ML/AI training, evaluation, experiments, and data workflows.
Do not start statistics exam prep by memorizing formulas. Start with the sequence of data, probability, sampling, inference, regression, then connect those ideas to model evaluation, A/B testing, and uncertainty in ML/AI.
The core of statistics is judgment: describe data, estimate unknowns, compare differences, inspect associations, and make decisions under uncertainty.
Data type determines the statistical tools you can use. Start with categorical, numeric, count, and time-ordered data, then choose summaries that fit the question.
Probability problems are often hard because the viewpoint changes. Define events first, then distinguish conditioning, independence, mutual exclusivity, and Bayes' rule.
Random variables turn uncertain outcomes into numbers. PMF, PDF, and CDF then let you compute discrete probabilities, continuous interval probabilities, thresholds, and model-score distributions.
Distributions are names for data-generating situations, not formula cards. Learn when Bernoulli, Binomial, Poisson, and Normal distributions fit a problem.
Expectation describes long-run center; variance describes fluctuation. This post computes E[X], E[X^2], and Var(X), then connects them to average loss and model stability.
Sampling makes sample statistics fluctuate, and standard error describes that fluctuation. This post separates SD, SE, sampling distributions, and CLT, then connects them to benchmark uncertainty.
A confidence interval puts a point estimate back inside sampling fluctuation. Computing bounds is only the first step; you also need to explain standard error, critical values, and coverage.
A hypothesis test is a decision process under uncertainty: write H0/H1, choose alpha, compute a test statistic and p-value, then decide whether the data is strong enough to challenge H0.
Two-group comparisons start by classifying the outcome and the design: numeric or binary, independent or paired. That choice determines the standard error, test statistic, and conclusion.
Chi-square tests compare observed counts with expected counts. First decide whether the problem is goodness-of-fit for one categorical variable or independence for two categorical variables.
ANOVA first checks whether three or more group means differ overall, so you do not inflate false-positive risk by running many pairwise t tests.
An estimator is a rule for using samples to infer a population parameter. To judge whether it is good, look at bias, variance, and MSE together.
Simple linear regression uses one X to describe the average change in Y. Slope, intercept, residuals, and squared error form the smallest supervised learning model.
A regression table is not a p-value list: coef, SE, t, F, and R-squared answer effect size, uncertainty, single-coefficient tests, overall model signal, and in-sample explanation.
Logistic regression connects a linear score to a probability between 0 and 1. Understanding odds, log odds, and odds ratios prevents wrong coefficient interpretations.
Joint PMF problems require listing every cell. Marginalization, conditional probability, and variable transformations are all sums or regroupings of the original cells.
Past papers train question-analysis discipline, not fortune-telling. Each problem should return to data type, unknown quantity, statistical tool, calculation path, and contextual conclusion.
At the final review stage, train problem recognition: identify data type, unknown quantity, and decision goal before choosing a formula and writing a contextual conclusion.
A sample is the data, a statistic is a function of the sample, and a sampling distribution is the distribution of that statistic under repeated sampling.
A sampling distribution describes how a statistic fluctuates under repeated sampling. Means, proportions, and variances each connect to common distributions used in intervals and tests.
Bias checks whether an estimator is centered correctly, variance checks sampling fluctuation, MSE combines both, and consistency asks whether the estimator approaches truth as sample size grows.
Method of Moments matches sample moments to theoretical population moments, then solves for parameters. It is not always the most efficient method, but it builds the first intuition for parameter estimation.
MLE fixes the observed data and compares which parameter values make that data most plausible; log likelihood turns products into sums and connects directly to negative log loss.
Fisher information uses likelihood curvature to measure how well the data locate a parameter; larger information usually means a smaller standard error for the MLE.
The likelihood-ratio test compares the log likelihood of a restricted model with a full model; the usual chi-square reference only makes sense under nested-model and approximation conditions.
The Neyman-Pearson view treats a test as a decision rule: under a fixed Type I error rate alpha, choose the rejection region with the highest power.
A confidence interval is built by defining the target estimate, describing its sampling error, and choosing a rule that turns uncertainty into a range.
Large-sample normal approximation describes the behavior of estimators, not raw data. It is useful, but dependence, boundaries, and distribution shift can make it unreliable.
The delta method transfers uncertainty through a smooth function: the local derivative expands or shrinks the estimator's original standard error.
Bootstrap estimates uncertainty by resampling from the observed sample with replacement, rebuilding many sample-like datasets, and watching the statistic fluctuate.
Bayesian inference updates uncertainty about an unknown parameter by combining prior belief with the likelihood from observed data, producing a posterior distribution.
MAP maximizes the posterior. After taking logs, the prior becomes a penalty term, which connects Bayesian estimation to L1, L2, and regularized ML objectives.
The inference map starts with the question type: point estimate, uncertainty interval, decision test, likelihood model comparison, Bayesian update, or resampling.
OLS is a useful baseline, but coefficient interpretation, inference, prediction, and diagnosis depend on assumptions about linearity, errors, independence, and variance.
Logistic regression estimates probabilities first. Classification decisions come later, when thresholds turn those probabilities into actions under real error costs.
A generalized linear model starts from the response type, chooses a suitable distribution, and uses a link function to connect the mean to a linear predictor.
Model diagnostics turn fitted errors into evidence: residual patterns, outliers, leverage, and influential points reveal how a model fails.
Variable selection is not only about choosing predictors. It is about avoiding noisy training-set wins that do not generalize.
Regularization adds a preference against extreme parameters. Ridge, Lasso, and weight decay trade some training fit for a model that generalizes more reliably.
Nonparametric methods are not assumption-free. They relax fixed distributional forms, often gaining flexibility while paying in efficiency, interpretation, or overfitting risk.
Experimental design decides whether a result can be interpreted. Randomization, control, blocking, replication, blinding, and pre-specified outcomes give inference a usable foundation.
A/B testing turns a product change into an estimate with uncertainty. A useful report covers effect size, confidence, guardrails, randomization, and launch risk.
Causal inference separates prediction from effect. A model can predict who will buy without proving that an intervention will make them buy.
Matching and weighting do not turn observational data into a true experiment. They try to make treatment and control comparable on observed variables.
Time-series data have order. Random splits can leak future information into training and make forecasting or monitoring results look better than they are.
Multivariate analysis looks at features together. Covariance, correlation, and PCA reveal shared directions that univariate summaries miss.
Missing data can change representativeness, bias estimates, and mislead ML systems. The first question is why the data are missing.
Monte Carlo repeats a data-generating process many times so sampling variation, power, coverage, and evaluation instability become visible.
A reproducible workflow preserves the evidence chain from data to conclusion. Results need data versions, code, seeds, environment, metrics, and raw outputs.
A useful ML/AI evaluation report turns statistical evidence into a decision: ship, stage, roll back, or run more experiments.
The series does not finish all of statistics. It gives beginners a working map for exams, ML/AI evaluation, causality, Bayesian thinking, time series, and mathematical statistics.