Know Statistics Through Its Branches

Statistics begins by describing observed data, uses probability to model uncertainty, learns from samples, builds models of patterns and time, and supports decisions. A branch is a tool family, not a separate subject.

A probability distribution assigns probability to possible values. Discrete models describe countable outcomes; continuous models use density, whose area over an interval gives probability.

Expectation locates a long-run center, variance measures spread, Markov and Chebyshev bounds limit tail probability, and laws of large numbers explain when sample averages approach a target.

Define the target population, choose a sampling frame and probability design, record inclusion probabilities, then weight observations when selection chances differ. Representation is created by design, not by sample size alone.

A statistic estimates an unknown parameter. Its sampling distribution supplies standard error, confidence intervals, and hypothesis tests that quantify what repeated valid samples could change.

Bayesian inference combines a prior distribution with the likelihood of observed data to form a posterior distribution. Posterior predictions and credible intervals express uncertainty after the update.

Bootstrap resampling repeatedly draws samples with replacement from the observed data, recomputes the statistic, and uses the resulting empirical distribution to estimate bias, standard error, or an interval.

Regression models a conditional outcome from predictors. Linear, multiple, logistic, count, and mixed-effects models answer different questions because outcome type and dependence structure differ.

Machine learning maps features to predictions by minimizing a loss. Train, validation, and test roles must be separated; performance on unseen data, not training fit, measures generalization.

Time-series data can contain trend, seasonality, autocorrelation, and shocks. Models such as AR, MA, ARIMA, and state-space systems use temporal order rather than treating observations as exchangeable.

A stochastic process is a collection of random variables indexed by time or space. Its state, transition rule, dependence, and sample paths describe both what can occur and how trajectories unfold.

Brownian motion has continuous paths, independent Gaussian increments, and variance proportional to elapsed time. Adding drift and diffusion produces stochastic differential models used for noisy evolving systems.

A Markov chain moves among states with transition probabilities depending on the current state. A transition matrix describes one-step motion; powers describe multiple steps and may approach a stationary distribution.

Monte Carlo generates random inputs, computes an output for each run, and averages the results. Accuracy improves at the familiar one-over-square-root-of-runs rate when variance is finite.

Markov chain Monte Carlo proposes moves, accepts or rejects them so the chain targets a desired distribution, discards warm-up, and checks mixing and effective sample size before using posterior summaries.

Survival analysis models event time while accounting for censoring. The survival function gives the chance of remaining event-free; the hazard describes the instantaneous event rate among those still at risk.

Arrival rate, service rate, server count, and variability determine congestion. When stable, Little's Law links average number in a system, throughput, and average time: L equals lambda times W.

Self-information measures surprise, entropy averages surprise, cross-entropy scores predictive distributions, KL divergence measures directional mismatch, and mutual information measures shared dependence.

Decision theory lists actions, uncertain states, losses or utilities, and constraints. An optimal action minimizes expected loss under a probability model; sensitivity analysis checks whether the decision is robust.

An agent observes a state, chooses an action, receives a reward, and transitions to a new state. A policy balances exploration and exploitation to maximize expected cumulative reward, connecting models to sequential decisions.