Family (statistics)

A family of probability distributions is a collection whose members share a common mathematical form and differ in parameter values.

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Parameter: A fixed value that specifies a particular model or distribution within a broader class. Different parameter values select different members of the family.
Exponential family: A class of distributions whose densities can be expressed using a particular exponential form and finite-dimensional sufficient statistics. It is a structured family with especially useful properties for inference.
Maximum likelihood estimation: A method that chooses parameter values maximizing the likelihood of observed data. It estimates which family member best accounts for a sample.
Probability distribution: A mathematical description of the possible outcomes of a random variable and their probabilities. A single distribution is one member; a family contains multiple parameterized members.
Probability density function: A function describing the relative likelihood of values for a continuous random variable. A shared density form often defines how family members relate.
Location-scale family: A family formed by shifting and rescaling a fixed base distribution. Its parameters alter location and spread without changing the standardized shape.
Bayesian inference: Inference that combines a probability model for data with a prior distribution over unknown quantities. A likelihood family supplies the data model used to update parameter beliefs.
Parametric statistics: Statistical methods that assume data follow a model indexed by a finite number of parameters. Families commonly underpin parametric methods, but the terms describe different things.
Probability mass function: A function assigning probabilities to the possible values of a discrete random variable. Discrete families share a mass-function form as their parameters vary.
Reparameterization: A change in the variables used to express a mathematical or statistical model. Equivalent parameterizations can describe the same members of a family.