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In my lecture note, a real-valued random variable is defined as follows:

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Then I have an exercise:

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Actually, my lecture notes do not have a definition of a random variable generated by other ones.

Since $X$ is a real-valued random variable, $X$ is a measurable function from $(\Omega_1, \mathcal{A}_1, \mathbb{P}_1)$ to $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$. Similarly, $Y$ is a measurable function from $(\Omega_2, \mathcal{A}_2, \mathbb{P}_2)$ to $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$. Because $Z$ is real-valued random variable, it is certainly a measurable function from some probability space $(\Omega, \mathcal{A}, \mathbb{P})$ to $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$.

My confusion:

  1. What does $Z$ actually mean by the notation $Z =X/Y$?

  2. What is $(\Omega, \mathcal{A}, \mathbb{P})$ when we already have $(\Omega_1, \mathcal{A}_1, \mathbb{P}_1)$ and $(\Omega_2, \mathcal{A}_2, \mathbb{P}_2)$?

  3. For $\omega \in \Omega$, what is $Z(\omega)$ when we already have $X(\omega_1), Y(\omega_2)$ for all $\omega_1 \in \Omega_1, \omega_2 \in \Omega_2$?

Thank you so much for your help!

Akira
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    In an intuitive sense, $Z = X/Y$ means that you sample $X$ (from whatever distribution $X$ follows), then sample $Y$, and $Z$ will be equal to $X/Y$. – glowstonetrees Sep 28 '19 at 07:06
  • Hi @glowstonetrees, I'm just exposed to the probability theory, I actually don't understand what you mean by "sample $X$" and "sample $Y$"? Could you please elaborate more? – Akira Sep 28 '19 at 07:08
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    Why not edit the previous post instead of asking the same thing again? – StubbornAtom Sep 28 '19 at 07:11
  • @StubbornAtom Because there are many comments below my question, I'm afraid that my edit will distort the context of such comments. – Akira Sep 28 '19 at 07:14
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    Since you add more context here (after the comments on the previous question), it would have been an improvement to that post. – StubbornAtom Sep 28 '19 at 07:23
  • @StubbornAtom Could you please vote to close the other one? – Akira Sep 28 '19 at 07:24

1 Answers1

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In the exercise it is assumed that $X$ adn $Y$ are defined on the same probability space. Otherwise we cannot attach any meaning to $\frac X Y$. When they are defined on the same probability space $Z=\frac X Y$ is defined by $Z(\omega)=\frac {X(\omega)} {Y(\omega)}$

Kavi Rama Murthy
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