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For a pair of random variables $Y$ and $Z$, is it possible that their ratio $X:=\frac{Y}{Z}$ is (exactly, not asymptotically) normally distributed?
If so, could you offer an example of the distributions of $Y$ and $Z$ and the relationship between them (besides the obvious case where $Y$ is normal and $Z$ is a constant, as suggested by @gunes)?

P.S. A special case of my question has been answered here: What Ratio of Independent Distributions gives a Normal Distribution?. My question is more general than that.

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  • $\begingroup$ Here is a related question: stats.stackexchange.com/questions/162483/… with a useful comment just under it. It seems the trivial case is the only case for independent RVs. $\endgroup$ – gunes Apr 3 at 16:58
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    $\begingroup$ @Gunes there are slightly less trivial cases. One is the ratio of a half-normal and an independent Rademacher variable. Apart from requiring the variables be continuous (and obviously one of them must be), stats.stackexchange.com/questions/121752/… is exactly the same question. $\endgroup$ – whuber Apr 3 at 17:31
  • $\begingroup$ It's a very good trick, i.e. Y half-normal and Z being 1,-1 with 0.5 prob. $\endgroup$ – gunes Apr 3 at 20:36
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    $\begingroup$ You probably want to say independent RVs to exclude things like $Z^2/Z$. $\endgroup$ – Hasse1987 Apr 4 at 1:39
  • $\begingroup$ @Hasse1987, thanks. I wanted the general case, including dependent $Y$ and $Z$. Once I got some good examples on that, the remaning interest is of course on independent cases. But that was not the sole focus of the question before I got the examples of the dependent cases. $\endgroup$ – Richard Hardy Apr 4 at 10:58
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A trivial case: Let $Y$ be a normal RV, and $Z$ be a constant RV, then $X$ is going to be normally distributed.

Another one: let $A,B$ normal RVs, and $C=A/B,D=1/B$ are two other RVs that belonging to Cauchy and Reciprocal Normal Distributions. Their ratio will be $C/D=A$ normally distributed.

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    $\begingroup$ It would be more interesting with some less obvious examples ... $\endgroup$ – kjetil b halvorsen Apr 3 at 16:36
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    $\begingroup$ Thank you! Techincally you are right. I did not exclude the constant RV case explicitly as I thought it was too obvious, but obviously it was not. I will edit my post. (P.S. I did not downvote your answer.) $\endgroup$ – Richard Hardy Apr 3 at 16:37
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    $\begingroup$ couldn't understand why I'm downvoted. (thank you, I guessed that you didn't, it was a general question). It'd be interesting if we have two Rvs that are independent and their ratio is normally distributed. Otherwise, we can define any dependence relation we want just as in the example case. $\endgroup$ – gunes Apr 3 at 16:38
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    $\begingroup$ The latter idea is a nice one. The case of independent $Y$ and $Z$ is even more interesting, but there is less hope there, I guess. $\endgroup$ – Richard Hardy Apr 3 at 17:36
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    $\begingroup$ The answer would be improved if the contrived trivial case were removed, and the substance left. $\endgroup$ – wolfies Apr 6 at 15:01

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