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I recall that the ratio of 2 independent standard normal random variables follows a Cauchy distribution, but I don't recall if there is a proof for this.

AdamO
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    There is in the wiki page: https://en.wikipedia.org/wiki/Ratio_distribution#/Uncorrelated_central_normal_ratio – gunes Mar 15 '21 at 21:15
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    https://stats.stackexchange.com/questions/52906/student-t-as-mixture-of-gaussian is a closely related thread showing how all Student t distributions can be obtained as scale mixtures of Normals. Also, if you go through Fisher's demonstration at https://stats.stackexchange.com/a/151969/919 for the simplest case $s=1$ you will see the Cauchy emerge in a natural geometric way from a ratio of iid Normals. At that point it's worth re-reading Douglas Zare's [nice characterization of the Cauchy distribution](https://stats.stackexchange.com/a/36037/919). – whuber Mar 15 '21 at 21:41

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