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My objective is to find out the distribution of $A/B$ given $A \sim N(a,b); B \sim N(c,d)$.

I set $Z_1$ equal to $\frac{(A-a)}{\sqrt{b}}$ and $Z_2$ equals $\frac{(B-c)}{\sqrt{d}}$ such that $Z_1$ and $Z_2$ are both $N(0,1)$.

Then I let $U=\frac{Z_1}{Z_2}$ and try to find out $g(U)$ (standard Cauchy distribution).

Could someone please let me know whether I am on the right track or not?

Silverfish
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John_Meri877
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  • Thanks for your reply. But I would want to know how to find out the joint distribution of A and B? – John_Meri877 Nov 21 '15 at 20:17
  • The distribution of $A/B$ is different than the joint distribution of $A$ and $B$. Which do you want to find? – Matt Brems Nov 21 '15 at 20:21
  • @MattBrems I am interested to know the distribution(or behavior) of A/B. – John_Meri877 Nov 21 '15 at 20:24
  • Are A and B independent? – Matt Brems Nov 21 '15 at 20:34
  • @MattBrems Yes they are – John_Meri877 Nov 21 '15 at 20:35
  • Then, yes, you're on the right track. Define C = A/B. Then C is Cauchy. If a=c=0 and b=d=1 (as you defined them above), then C ~ Cauchy(0,1). If a and c aren't 0 or b and d aren't 1, then the parameters will be different. If you standardize them as you did above, then U ~ Cauchy(0,1). – Matt Brems Nov 21 '15 at 20:40
  • The difficulty is that $A/B$ is not a transform of $Z_1/Z_2$! – Xi'an Nov 21 '15 at 20:58
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    Duplicate of http://stats.stackexchange.com/q/86178/6633 where Moderator Glen_b's answer includes a link to a .pdf file which contains the full answer for the distribution of $A/B$ where $A$ and $B$ are jointly normal random variables with arbitrary means and variances. The answer is _not_ always Cauchy (only) as Matt Brems's comment seems to imply but is related to the Cauchy density. – Dilip Sarwate Nov 21 '15 at 21:56
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    @DilipSarwate That is an excellent spot. That was a side-issue in that (rather confused) question though. For example, if at a later date somebody wanted to include a full derivation then it would arguably belong more at this question than at that one. And somebody searching for an answer to the "normal over normal" question would find the other issues on that thread a distraction. Arguably that thread might better be closed as a duplicate of this one, because of the "untidiness" there (that arose from asking a question from a partially false premise). – Silverfish Nov 21 '15 at 22:12
  • I can't quite face following it up at this time of night, but [Wikipedia](https://en.wikipedia.org/wiki/Cauchy_distribution#Transformation_properties) looks like it might be helpful here. – Creosote Nov 21 '15 at 23:14
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    I don't believe this site contains a fully worked answer to this general question, but it does answer special cases and provides extensive references to answers, including http://stats.stackexchange.com/questions/33050, http://stats.stackexchange.com/questions/3640, http://stats.stackexchange.com/questions/175445, http://stats.stackexchange.com/questions/47009, http://stats.stackexchange.com/questions/162483, http://stats.stackexchange.com/questions/28233. Since $Z_1/Z_2$ has a distinctly different distribution than $A/B$, using the $Z_i$ is of little help. – whuber Nov 22 '15 at 16:42
  • See also https://stats.stackexchange.com/questions/175445/distribution-of-ratio-of-correlated-normals – kjetil b halvorsen Nov 20 '19 at 15:05

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