Probability Distribution

Distribution parameters

Definition

A parameter $\theta$ is a quantity which characterizes the distribution $f(x,\theta )$.

In the presence of multiple parameters we say that $\theta$ is vector containing these parameters. An example would be $N(\mu ,{\sigma }^2)\to N({\theta }_1,{\theta }_2)$.

Parameter space $\Omega$ is set of all possible values for the $\theta$.

Parametric family of distributions

Let $\theta \in \Omega$. Then such a family is defined by $\{f(x,\theta ):\theta \in \Omega \}$. Any member of this family of distribution is denoted by $f(x,\theta )$ given that $\theta \in \Omega$.

In other words to each $\theta \in \Omega$ there exists a pdf.

Normal distribution as a parametric family

Observe that

$$f(x,\theta )=\frac{1}{\sqrt{2\pi {\theta }_2^2}}\exp \left(-{\left(\frac{x-{\theta }_1}{{\theta }_2}\right)}^2\right)$$ $$\Omega =dom(\mu )\times dom({\sigma }^2)=\mathbf{R}\times (0,\infty )$$

A parametric family of distributions would be $\{N(\theta ,1):\theta \in \Omega \}$ given that $\Omega =\mathbf{R}$.