And yet you still can't name a single real system (not a mathematical abstraction) which is non-random by your definition.
I have, but you insist that I am only referring to "mathematical abstractions".
And by the way, how about a link to one of those "experts in probability theory" using your definition?
Try any textbook or journal article ever published in probability theory. Of course, that assumes the one knows how a probability distribution is defined in probability theory:
Basically, the axiomatization of probability theory begins with the creation of a mathematical space called a
probability space designated by the ordered triple [latex]$(\Omega, \mathcal{F}, \mathbb{P})$[/latex]. [latex]$\Omega$[/latex] is a set, which, for our purposes, is almost any collection of objects. If [latex]$\Omega[/latex] is finite or countably infinite, the probability space is called discrete. If [latex]$\Omega[/latex] is uncountably infinite, the probability space is called continuous. [latex]$\mathcal{F}$[/latex] is a family of subsets of [latex]$\Omega$[/latex] called a [latex]$\sigma$[/latex]-algebra whose members share certain attributes and are known as "events". In particular, a [latex]$\sigma$[/latex]-algebra is closed under complementation (i.e., the "opposite" of every event is also an event) and countably infinite unions (i.e., a collection of events is also an event). [latex]$\mathbb{P}$[/latex] is a function called a probability measure that maps each event in [latex]$\mathcal{F}$[/latex] to the interval [0,1] in such a way that [latex]$\mathbb{P}(\Omega)=1$[/latex] and the probability of the union of mutually exclusive events is the sum of their respective probabilities.
A
random variable is, by definition, a function, [latex]$X$[/latex] from the probability space [latex]$(\Omega, \mathcal{F}, \mathbb{P})$[/latex] to another measurable space, most often [latex]$(\mathbb{R}, \mathcal{B}(\mathbb{R})$[/latex] , the real numbers and the Borel [latex]$\sigma$[/latex]-algebra on [latex]$\mathbb{R}$[/latex] (i.e., the smallest [latex]$\sigma$[/latex]-algebra form by the open intervals of [latex]$\mathbb{R}$[/latex]). The random variable defines a probability measure, known as a pushforward probability measure, on the measurable space [latex]$(\mathbb{R}, \mathcal{B}(\mathbb{R})$[/latex] such that [latex]$(X_*(\mathbb{P}))(B)=\mathbb{P}(X^{-1}(B))[/latex]. In other words, that is the pushforward probability measure of [latex]$B\in\mathcal{B}(\mathbb{R})$[/latex] equals the probability measure of [latex] [latex]X^{-1}(B)\in\mathcal{F}[/latex], the preimage of [latex]$B$[/latex] in [latex]$\mathcal{F}[/latex] (i.e., [latex]$\{A\subset\Omega}\}\in\mathcal{F}$[/latex] that maps to [latex]$\{B\subset\mathbb{R}\}\in\mathcal{B}(\mathbb{R})$[/latex]). This pushforward measure is called a
probability distirbution.
A
stochastic process is a family of random variables [latex]$\{X_{t}\}_{T\in{t}}$[/latex] for an index set [latex]$T$[/latex] If [latex]$\Omega[/latex] is finite or countably infinite, the stochastic process is called discrete. If [latex]$\Omega[/latex] is uncountably infinite, the stochastic process is called continuous.
