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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
    Commented Apr 3, 2020 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 ♦
    Commented Apr 3, 2020 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
    Commented Apr 3, 2020 at 20:36
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    $\begingroup$ You probably want to say independent RVs to exclude things like $Z^2/Z$. $\endgroup$
    – Hasse1987
    Commented Apr 4, 2020 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
    Commented Apr 4, 2020 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 ♦
    Commented Apr 3, 2020 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
    Commented Apr 3, 2020 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
    Commented Apr 3, 2020 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
    Commented Apr 3, 2020 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
    Commented Apr 6, 2020 at 15:01

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