I just want to confirm a convention in probability that whenever we talk about a random variable, we assume that the codomain is $\mathbb{R}$.
That is, whenever $A$ is a random variable, $\text{codomain}(A) = \mathbb{R}$
Is this correct?
I just feel like this is a very unfortunate omission in most elementary probability textbooks. The textbook authors then go on $5$ more chapters talking about these scalar-valued random variables.
Because in real-life, almost all random experiments include a large set of variables, hence almost all real-life random experiments involve "vectorized random variable" or "stacked random variable". At least this is the case when you grab any dataset from a repository.
Hence to beginners this may seem that the whole idea of random variable is useless or not practical. I mean just look at the most elementary data from Kaggle, you would at least have 2 entries (height,weight), (age, degree). Hence almost all random variable on Kaggle are vectorized random variables.