概率與統計：不確定性的科學

This lesson introduces formal probability models as a rigorous framework to replace subjective intuition, highlighting the relative frequency interpretation and the Law of Large Numbers. Students learn to apply these mathematical structures to quantify uncertainty and manage risk in complex, real-world scenarios where human cognitive biases often fail.

This lesson introduces random variables as deterministic functions that map sample space outcomes to real numbers, providing a quantitative framework for probability. Students learn to utilize indicator functions, understand probability distributions, and apply the continuity of probability to analyze complex events.

This lesson introduces mathematical expectation as the long-run average of a random variable and explores its core properties, including linearity, monotonicity, and independence. Students learn to apply the Law of the Unconscious Statistician (LOTUS) to efficiently calculate the expected values of transformed variables without needing to derive their specific probability distributions.

This lesson introduces sampling distributions as the probability laws governing statistics, which are functions of independent and identically distributed (i.i.d.) random variables. Students learn to derive exact distributions for small sample sizes and explore how these concepts bridge the gap between raw data and statistical inference.

This lesson introduces statistical inference as the formal process of using sample data to estimate the underlying probability distributions and mechanics of a system. It emphasizes that inference is necessary to distinguish between inherent random variation and structural uncertainty, allowing researchers to make robust predictions beyond simple descriptive summaries.

This lesson explores likelihood-based inference, focusing on how the likelihood function quantifies the support for different parameter values given observed data. Students learn to use log-likelihoods for computational efficiency, apply the Central Limit Theorem for asymptotic inference, and utilize Fisher Information to measure the precision of statistical estimates.

This lesson introduces the Bayesian paradigm, which treats unknown parameters as random variables rather than fixed constants to allow for direct probability statements about them. Students learn to construct a complete Bayesian model by combining a sampling model with a prior distribution to update beliefs through the joint distribution.

This lesson explores the mathematical foundations of optimal statistical inference by defining the Mean Squared Error (MSE) as the sum of an estimator's variance and squared bias. Students learn how to minimize this error to identify the best estimators and understand the role of sufficiency and posterior means in decision theory.

This lesson introduces model checking as a critical validation step that ensures statistical inferences are grounded in reality rather than mathematical fiction. Students will learn to distinguish between parameter estimation and model validation, emphasizing that even the most precise calculations are meaningless if the underlying model assumptions do not accurately reflect the data-generating process.

This lesson defines a statistical relationship as any change in the conditional distribution of a response variable $Y$ when a predictor $X$ varies, moving beyond simple correlation to include shifts in mean, variance, or shape. It also emphasizes that establishing causality requires rigorous experimental design, such as blinding and blocking, to account for confounding variables and eliminate bias.

This lesson introduces stochastic processes as systems that evolve over time through probabilistic rather than deterministic rules, with a primary focus on the Simple Random Walk. Students learn to calculate path probabilities and expected values while exploring key concepts like the parity rule, Martingale fairness, and the foundational mechanics of the Gambler’s Ruin model.