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I am looking to find the pdf of the ratio $Z = \frac{X}{Y}$, where $X \sim$ Gaussian and $Y \sim$ Chi-squared are indepedent random variables.

A reference is good enough if you do not want to write down the complete solution.

wolfies
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GAA
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    How about this https://stats.stackexchange.com/questions/151854/a-normal-divided-by-the-sqrt-chi2s-s-gives-you-a-t-distribution-proof – user158565 Dec 08 '18 at 17:10
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    @user158565 that's normal divided by the square root of a scaled chi-square; which is not what this question asks. You might guess that OP has a mistake in their question (probably correctly) but it's best to make sure of it. – Glen_b Dec 08 '18 at 17:13
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    @GAA your title question asks about dividing distributions, but you seem to actually be asking about dividing random variables; please clarify your title so it doesn't imply you're taking a ratio of distributions, but of random variables. – Glen_b Dec 08 '18 at 17:14
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    Since GAA is new, I have edited the question to reflect what we think he is asking. GAA, can you please also clarify whether your Gaussian is standard Normal, or general Normal? – wolfies Dec 08 '18 at 17:58
  • This question has already been asked on this forum. I cannot trace it though. – Xi'an Dec 08 '18 at 18:38
  • @GAA please address the issue of whether you really do intend to divide by a chi-squared or by its square root (possibly scaled), since the second thing commonly arises. – Glen_b Dec 09 '18 at 02:29
  • Thank you very much for your time and interest. X is a random variable follows the Gaussian distribution with mean zero and variance sigma ( i.e., standard Normal ) and Y is a random variable follows the Chi-squared distribution. What is the distribution of Z=X/Y – GAA Dec 14 '18 at 22:35
  • @Glen_b I have come across the same problem. That is, how can I determine the distribution of X/Y for X standard normal and Y chi squared? Or, if i understand correctly, the distribution of X/Y where X is t-distributed and Y is square root of chi squared? I'm having trouble finding a reference/methodology – curious_dan Nov 15 '20 at 16:53
  • (i) Which is your problem -- to divide by a chi-squared, or its square root? (ii) Is the denominator independent of the numerator? – Glen_b Nov 16 '20 at 01:54
  • @Glen_b sorry I didn't mean to confuse the situation. Let's use the following formulation: X standard normal divided by Y chi-squared (no square root). Suppose they are independent. Is this equivalent to asking what is the distribution of W/sqrt(Y) where W is t-distributed and sqrt(Y) is square root of chi-squared, with independence? – curious_dan Nov 16 '20 at 16:42

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