Is responses in statistics the equivalent to random variables in probability?

Question

this post says

The focus of this class is multivariate analysis of discrete data. The modern statistical inference has many approaches/models for discrete data. We will learn the basic principles of statistical methods and discuss issues relevant for the analysis of Poisson counts of some discrete distribution, cross-classified table of counts, (i.e., contingency tables), binary responses such as success/failure records, questionnaire items, judge's ratings, etc. Our goal is to build a sound foundation that will then allow you to more easily explore and learn many other relevant methods that are being used to analyze real life data. This will be done roughly at the introductory level of the first part of the required textbook by A. Agresti (2013), which covers a superset of A. Agresti (2007)

in which, is responses here (statistics) the equivalent to random variables in probability

another page in that site says

Discretely measured responses can be:

1. Nominal (unordered) variables, e.g., gender, ethnic background, religious or political affiliation
2. Ordinal (ordered) variables, e.g., grade levels, income levels, school grades
3. Discrete interval variables with only a few values, e.g., number of times married
4. Continuous variables grouped into small number of categories, e.g., income grouped into subsets, blood pressure levels (normal, high-normal etc)

We we learn and evaluate mostly parametric models for these responses.

are variables and responses interchangeable here?

Answer 1

Whereas the term variable can be understood broadly, I believe here the author is making distinction between variable as a number (integer or real) and response as a random variable from probability theory (as a variable which can take values from some set $X$, which is a part of measurable space $(X,\Sigma)$).

For example, the variable “age” is a continuous variable from $0$ to $\infty$. However, the data and/or statistical model may only distinguish between children (0-11 y.o.), teens (12-19 y.o.), adults (20-60 y.o.) and elders (60-$\infty$ y.o.) groups. So the response can take the value of one of those words.

You are right, that from the perspective of probability theory, the response is a random variable on some uncommon probability space. However, if one wants to be able to add and multiply their variables, they would like to come up with other term (like response) for things with just probability space and no obvious algebra.