Definition (Parametric family)

Ley $y$ be the data vector and $\theta$ be the vector of unknowns. Ley $p(y,\theta)$ be the full joint pdf of $(y,\theta)$. Let $p(y)$ and $p(\theta)$ denote the marginal pdf of $y$ and $\theta$ respectively. Let $p(y\mid\theta)$ denote the conditional pdf of $y$ given $\theta$. A parametric family is a collection of conditional pdf/pmfs $p(y\mid\theta)$ indexed by $\theta$. The index $\theta$ is called the parameter. We define $$\mathscr{P}=\{ p(y\mid\theta):\theta \in\Omega \}$$where $\Omega$ is called the parameter space.