Statistical Models

Def Statistical Model (Family)
A statistical model (family) is a collection of probability distributions on some sample space. We assume that the collection, $\mathcal{P}$, is indexed by some set $\Theta$. The set $\Theta$ is called the parameter space. For each $\theta \in \Theta$, let $F_{\theta }$ denote the corresponding member of the collection. Then a statistical model can be written as $\mathcal{P}=\{F_{\theta }\mid \theta \in \Theta \}$.

Def Parametric Model (Family)
A statistical model (family) is parametric if $\Theta \subseteq {\mathbb{R}}^k$ for some positive integer $k$.

Probability Distributions

The Basic Distributions

Uniform Distribution

The Exponential Family

• Binomial Distribution
• Beta Distribution
• Gamma Distribution
• Pareto Distribution
• Exponential Distribution
• Poisson Distribution