Setup
I'm taking a course on probability and our professor has defined random variables in a way that I'm not so used to, I'm used to the definition of a random variable X, being a function from the sample space into the real numbers. Instead we are given this:
And then we get random varaibles:
Usage
A sample definition using this notation is:
What I don't understand is what a system actually is. From what I can gather a system, is like some sort of function which maps into our sample space. But if that's the case than what would the domain of our system W be?
Another idea I had was that a system is just like a random variable, but it need not map out to the reals, instead just some arbitrary set, and then the random variable maps from that set to the reals.
My hunch is that they are trying to decouple the idea of the random event, and then a random variable which produces a real number from that event, but I'm not entirely sure. If anyone has seen this before and knows formally what a system is could they please explain?
Best Answer
At least the class still uses a sample space $\Omega$ and still talks about functions $g:\Omega\rightarrow\mathbb{R}$ (putting details about sigma algebras aside, these functions $g$ are what everyone else would call random variables).
I think your class wants to view $W$ as a "factory" (or piece of software) that, when repeatedly run, produces (independent) outcomes. It wants to use $W$ as the name of the software, and then use $w_1, w_2, w_3$ as "realizations" from $W$. It defines $X=g(W)$ as a new piece of software that puts the output of the previous software through $g$. So then "realizations" of $X$ are $g(w_1), g(w_2), g(w_3)$.
This view is a significant departure from standard probability theory where there is a single sample space $\Omega$, and the outcome $\omega \in \Omega$ determines the values $X_1(\omega), X_2(\omega), X_3(\omega), ...$ for all random variables $\{X_1, X_2, X_3, \ldots\}$. One way to make this consistent is if you consider the product space $$\tilde{\Omega} = \underbrace{\Omega \times \Omega \times ... \times \Omega}_{\mbox{$n$ times}}$$ so a single outcome $\omega \in \tilde{\Omega}$ has the form $\omega = (\omega_1, \omega_2, ..., \omega_n)$ and you can define $$X_i(\omega_1, ..., \omega_n) = g(\omega_i) \quad \forall \omega\in \tilde{\Omega}$$
The trouble with this is that it implicitly assumes all "trials" or "samples" are always independent. You should ask your professor how to handle cases when the $X_1, X_2, ..., X_n$ random variables can be correlated.
Regarding standard definitions I would use this:
A measurable space is a pair $(\Omega, \mathcal{F})$ where $\Omega$ is a nonempty set and $\mathcal{F}$ is a sigma algebra on $\Omega$.
A probability space is a triplet $(\Omega, \mathcal{F}, P)$ where $(\Omega, \mathcal{F})$ is a measurable space and $P:\mathcal{F}\rightarrow[0,1]$ a function that satisfies the 3 axioms of probability.
Now suppose we have a probability space $(\Omega, \mathcal{F}, P)$:
A random variable is a function $X:\Omega\rightarrow\mathbb{R}$ that satisfies $$ \{\omega \in \Omega : X(\omega)\leq x\} \in \mathcal{F} \quad \forall x \in \mathbb{R}$$ Equivalently, it can be shown that $X:\Omega\rightarrow\mathbb{R}$ is a random variable if and only if $$ \{\omega \in \Omega : X(\omega) \in B\} \in \mathcal{F} \quad \forall B \in \mathcal{B}(\mathbb{R})$$ where $\mathcal{B}(\mathbb{R})$ is the standard Borel sigma algebra on $\mathbb{R}$.
Given a measurable space $(V, \mathcal{G})$, a random element is a function $X:\Omega\rightarrow V$ that satisfies $$ \{\omega \in \Omega : X(\omega) \in B\} \in \mathcal{F}\quad \forall B \in \mathcal{G}$$
Under these definitions, a random variable is just a random element on the output measurable space $(V,\mathcal{G})=(\mathbb{R}, \mathcal{B}(\mathbb{R}))$.
For two measurable spaces $(V_1, \mathcal{G}_1)$ and $(V_2, \mathcal{G}_2)$ and two random elements $X:\Omega\rightarrow V_1$ and $Y:\Omega\rightarrow V_2$, we say $X, Y$ are independent if $$ P[\{X \in A\}\cap \{Y\in B\}]=P[X\in A]P[Y\in B] \quad \forall A \in \mathcal{G}_1, B \in \mathcal{G}_2$$ Some people call random elements "random variables." I suspect your last question on the definition 2.0.1 uses $\mathcal{X}=V_1$ and $\mathcal{Y}=V_2$, ignores the issue of sigma algebras, and if we assume $X, Y$ are real-valued then indeed you can assume $\mathcal{X}=\mathcal{Y}=\mathbb{R}$. Alternatively, perhaps $\mathcal{X}$ is the subset of $\mathbb{R}$ consisting of values that $X$ can take, that is, $\mathcal{X}=\{X(\omega): \omega \in \Omega\}$ (the image of the random variable). Some people use notation $S_X$ for this.