Cauchy Ratio Distribution
If two independent random variables, X and Y each follow a Cauchy distribution with median equal to zero and shape factor
then the ratio distribution for the random variable is
Interestingly, this distribution does not depend on and it should be noted that the result stated by Springer (p158 Question 4.6) is not correct. The ratio distribution is similar to but not the same as the product distribution of the random variable :
More generally, if two independent random variables X and Y each follow a Cauchy distribution with median equal to zero and shape factor and respectively, then:
1. The ratio distribution for the random variable is
2. The product distribution for the random variable is
The result for the ratio distribution can be obtained from the product distribution by replacing with
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