We first distinguish between the ideas of a functional relationship and statistical relationship. Consider a relation between two variables $x, y$.
A functional relationship deterministically (always) maps $x$ to one value of $y$.
Mathematically, we would denote this as:
$$ \mathbf y = \mathbf g(\mathbf x) $$
Sometimes, there are unexplained differences to the observer that can be attributed to natural variability or does not have a mathematical explanation. To handle this, we introduce the idea of a statistical relation:
In this case, we add a random variable, $\bm{\epsilon}$ to the output of the response function:
$$ \mathbf y = \mathbf g(\mathbf x) + \bm{\epsilon} $$
Implicit in this definition, is that $\mathbf y$ becomes a random
Representations of statistical relations