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In my Probability class, A random variable is a mapping $X: \Omega \to \mathbb{R}$, where $\Omega$ is equipped with a $\sigma$-field $\Sigma$ and a probability $\mathbb{P}$. I am having adapt this definition into applications.

For example, I have seen "the number of calls received during one day" is modeled by Poisson distribution. But what is the random variable $X$ defined here exactly? What is the domain of $X$? What is the corresponding $\sigma$-field and probability?

Furthermore, why do we normally not consider these underlying spaces for random variables? When do we actually care?

Partial T
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    This may be an underwhelming answer, but: Usually we do not care one bit about what the probability space $(\Omega,\mathcal F,\mathbb P)$ looks like. We only really care about things like the induced distribution $\mathbb P^X:=\mathbb P\circ X^{-1}$. While concepts like almost-sure convergence technically involve the underlying space $\Omega$, one usually doesn't need any "knowledge" about $\Omega$ to determine whether a sequence $(X_n)_{n\in\mathbb N}$ of RVs converges a.s. – Small Deviation Jul 05 '23 at 10:41
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    The domain of $X$ is the mysterious realm where the god of phone calls makes their decisions. – FShrike Jul 05 '23 at 10:53
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    When applying modern probability theory to real life it is typical to just assume that a sample space "exists", or to assume that this is a reasonable modelling assumption – FShrike Jul 05 '23 at 10:53
  • Often in the applications of probability theory the main interest is the distribution of a random variable or even a stochastic process. Building a random variable (or a sequence of random variables or even stochastic processes) with prescribed properties is not canonical, in the sense that there might be several probability spaces, say $(\Omega_\alpha,\mathscr{F}\alpha,\mathbb{P}\alpha)$, $\alpha\in\mathcal{A}$ and random variables $X_\alpha:(\Omega_\alpha, \mathscr{F}_\alpha)\rightarrow(\mathbb{R},\mathscr{B}(\mathbb{R}))$ that have the prescribed distribution. – Mittens Jul 05 '23 at 14:58
  • Related – Mittens Jul 05 '23 at 15:07
  • @SmallDeviation But if we do not care about what $(\Omega, \mathcal{F}, \mathbb{P})$ is, how does one talk about induced probabilities? Induced probability only makes sense when $\mathbb{P}$ is established, right? – Partial T Jul 06 '23 at 17:08
  • @PartialT One does not need much knowledge about $\Omega$ or $\mathbb P$ to define a distribution on the image space. For example, i can easily define a Gaussian distribution on $\mathbb R$ without ever needing to think of a probability space $\Omega$ or any probability measure $\mathbb P$ on $\Omega$. One then often simply says that "there exists some probability space $\Omega$" and a random variable $X$ defined on it, such that $X$ has e.g. a Gaussian distribution. – Small Deviation Jul 06 '23 at 18:10

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