parametric models and its notation

Question

I'm having a difficult time understanding parametric models and its notation.

In wikipedia:

"A parametric model is a collection of probability distributions such that each member of this collection, Pθ, is described by a finite-dimensional parameter θ."

So my understanding is that P is a set of different probability distributions (Pθ) of a collection of parameters (θ)?

Answer 1

@Math1000 is right. It's usually required that $\Theta \subset \mathcal{R}^d$ for a fixed $d>0$.

Another example is that $p_\theta = N(\mu, \sigma^2)$, with $\mu \in R $ and $\sigma^2 \in R^+$. Here $\theta = (\mu, \sigma^2)^T$, $\Theta=R \times R^+$. So returning back to the notation, $P=\{p_\theta:\theta \in \Theta\}$ is the collection of all 1-D normal distribution, which is a parametric model.

Hope it helps.