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I'm trying to solve this exercise:

Let $X,Y$ be real-valued random variables defined on the same probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and $\operatorname{supp} (X)= \operatorname{supp} (Y) = \mathbb N^*$. Find the support of $Z = X/Y$.

Here is the definition of support of a random variable in my lecture note:

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My attempt:

I guess that $\operatorname{supp} (Z)=\mathbb R_+$.

Assume that there exists $z \in \mathbb R_+$ s.t there is an open neighborhood $(a,b)$ of $z$ s.t $\mathbb P (Z \in (a,b)) = 0$. As such, for any $p,q \in \mathbb N^*$ such that $p/q \in (a,b)$, we have $\mathbb P (\{\omega \in \Omega \mid X(\omega) = p \,\,\text{and} \,\, Y(\omega) = q\}) = 0$.

I'm stuck at using the fact that $\operatorname{supp} (X)= \operatorname{supp} (Y) = \mathbb N^*$ to finish the proof.

Could you please shed me some light?

Akira
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  • What if $X=Y$ or $Y=aX $ with $a$ any constant...? – Shashi Sep 28 '19 at 12:59
  • @Shashi Then $Z= a$. Thank you for your comment. Could you please have a look at my closely related question https://math.stackexchange.com/questions/3372856/what-is-the-random-variable-z-by-the-notation-z-x-y? – Akira Sep 28 '19 at 13:17
  • that question has a perfect answer. What I wanted to tell you with my comment is that we can't tell the support of $Z$ with the information you provided... Does that make sense? – Shashi Sep 28 '19 at 13:25
  • Thank you so much @Shashi! This is a question from my lecture note. I retype it in LaTex verbatim. – Akira Sep 28 '19 at 13:27

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