Mathematical statistics

Mathematical statistics uses probability theory and mathematical methods to develop and analyze procedures for collecting, modeling, and interpreting data.

Probability theory: The mathematical study of random events, uncertainty, and the probabilities assigned to outcomes. Its rules turn assumptions about randomness into distributions that statistical procedures can analyze.
Statistical model: A mathematical representation of assumptions about how data are generated. Inference depends on how the model describes the data and its unknown parameters.
Bayesian statistics: An approach to statistical inference that combines probability models with prior distributions over unknown quantities. It incorporates prior distributions, unlike classical procedures that typically assess repeated-sampling behavior.
Regression analysis: A set of methods for modeling relationships between an outcome and one or more explanatory variables. Its estimates and tests apply mathematical inference to relationships among measured variables.
Jacob Bernoulli: A Swiss mathematician whose work established an early mathematical result on convergence of sample proportions. His law of large numbers helped connect repeated observations with stable statistical summaries.
Random variable: A function that assigns numerical values to outcomes of a random experiment. It represents sample measurements as quantities governed by probability distributions.
Maximum likelihood estimation: A method that estimates parameters by maximizing the likelihood of the observed data. It is a central way mathematical statistics turns a specified model into parameter estimates.
Frequentist inference: An approach that evaluates statistical procedures by their behavior over repeated samples. It supplies the repeated-sampling framework underlying many classical guarantees.
Analysis of variance: A method for comparing population means by partitioning observed variation into components. Its test statistics are derived from probability models for variation within and between groups.
Pierre-Simon Laplace: A French mathematician who developed probability methods and applied them to inference and astronomy. His work advanced probability-based estimation and the analysis of observational error.

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