Probabilistic modelling
Firstly, the following
It is assumed that $A$ is the constant random variable that only takes the value $a\in \mathbb{R}$, where $a$ is thought of as input.
does not really make sense. If something is assumed to be constant, then there is no need to model it as a random variable. Moreover if $A$ is going to model the input, it must vary from time to time. So let us assume that this is rather poorly written - $A$ is a random variable with some (maybe unknown) non-constant distribution. Maybe it should be discrete, or taking only two values? It might make sense.
Nextly, we have
Furthermore, $\mathbb{P}(B=b|A=a)=\mathcal{N}(b;3⋅e^a,1)$ and $\mathbb{P}(C=c|B=b)=\mathcal{N}(c;2⋅b^2,1)$, where $\mathcal{N}(x;\mu,1)$ denotes the probability density of a normal distribution with mean $\mu$ and variance $1$.
This is also weirdly written, and makes little sense. You have written
...on the left we have a number between 0 and 1 but on the right we have a number that can be larger...
You are write, we should not use $\mathbb{P}$ sign when the probability density function is really what we are referring to.
What the author probably wanted to say, was that conditionally on $A$, the distribution of $B$ is normal with given ($A$ dependent) parameters. Later on, after $(A, B$) have been sampled, the distribution of $C$ is also normal with the assigned ($B, A$ dependent) parameters.
The above is only my educated guess, as in the cited text it is written
$$\mathbb{P}(B=b|A=a)=\mathcal{N}(b;3⋅e^a,1),$$ \begin{equation} \mathbb{P}(C=c|B=b)=\mathcal{N}(c;2⋅b^2,1), \tag{1} \end{equation} and I assume that it should have been $$\mathbb{P}(B=b|A=a)=\mathcal{N}(b;3⋅e^a,1),$$ \begin{equation}\mathbb{P}(C=c|B=b, A=a)=\mathcal{N}(c;2⋅b^2,1). \tag{2} \end{equation} My guess is based on the fact, that as You say, we are actually going to model some process. As long as $(1)$ might be valuable piece of information, it does not define a unique model.
Moreover, this should be written in a more clear way though. For example:
$$B|A \sim \mathcal{N}(3\cdot e^A,1),$$ and $$C|B,A \sim \mathcal{N} (2\cdot B^2,1).$$
Note that this is exactly what You have written in Your interpretation
Given $a$, draw a number $b$ from a normally distribution with mean $3\cdot e^a$ and unit variance and then similarly draw $c$, the output, from a normally distribution with mean with mean $2\cdot b^2$ and unit variance.
I think it is OK. Saying it once again in different words:
- we start by sampling $A$,
- given $A$, we sample $B$ from $\mathcal{N}(3\cdot e^A,1)$,
- given $B, A$, we sample $C$ from $\mathcal{N} (2\cdot B^2,1).$
Such models - known as hierarchical models - are very important in modern statistics. Basically, Bayesian Statistics is devoted almost entirely to the study of such models.