Geodesic
A Family of Distributions with the same Ratio Property as Normal Distribution
Canadian mathematical bulletin, Tome 8 (1965) no. 5, pp. 631-636

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If U and V are independent random variables, both drawn from the Normal distribution, then it is known that the distribution of U/V follows the Cauchy law, i. e. has frequency function 1/π(1+x2). Conversely if U and V are independently drawn from the same distribution and U/V is known to follow the Cauchy law of distribution must U and V be necessarily drawn from a Normal distribution? This question has been considered by several authors ([3], [4], [5], [7]) who have obtained several examples to show that the ratio property above is not confined to the Normal distribution. 
Fox, Charles. A Family of Distributions with the same Ratio Property as Normal Distribution. Canadian mathematical bulletin, Tome 8 (1965) no. 5, pp. 631-636. doi: 10.4153/CMB-1965-046-3
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