Probability Foundations for Advanced Statistics
Why probability is the language of inference
From uncertainty to model-based reasoning
Advanced statistics treats uncertainty as a formal object. Random variables, events, and probability measures allow us to quantify variability, construct estimators, and justify inference under explicit assumptions.
Core viewpoint
A statistical model is a set of probability distributions indexed by parameters, and inference is the process of learning about those parameters from data.
Sigma-algebras and probability spaces
A probability space is , where is the sample space, is a sigma-algebra, and is a probability measure satisfying .
Foundational objects
| Object | Role |
|---|---|
| Set of all possible outcomes | |
| Collection of events to which probabilities are assigned | |
| Probability measure on events | |
| Random variable | Measurable map from outcomes to numbers |