What does it mean for a probability model to be parametric?

Question

Our course slides offer the following definition: "A parametric probabilistic model is a set of probability distributions indexed by a finite-dimensional parameter vector."

This description defines "parametric probabilistic model" in terms of "parameter", which isn't particularly helpful. But it brings to mind some vector of indices which tells you where to find a row (or rows) in a table.

It is not clear to me whether "parameter" in this sense is related to my CS understanding of a parameter (i.e., a "knob") or even if a model's parameters have something to do with its random variables (e.g., are the parameters fixed values of a random variable?). Furthermore, what does it mean to "parameterize" a model?

Answer 1

Maybe an example will help. The binomial distribution is a parametric model with two parameters: $n$ and $p$. That is, for each nonnegative integer $n$ and each $p \in [0,1]$ we have a probability distribution.

Answer 2

Parameters are not "fixed values of a random variable".

They are "knobs".

A parametric model involves a parametrized family of probability distributions. In the case of the normal distribution, one usually uses the expected value and the variance. In the case of the Poisson distribution, one usually uses the expected value, and that shows that it cannot coincide with values of the random variable, since those can only be integers.