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Concept# Ratio distribution

Summary

A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions.
Given two (usually independent) random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.
An example is the Cauchy distribution (also called the normal ratio distribution), which comes about as the ratio of two normally distributed variables with zero mean.
Two other distributions often used in test-statistics are also ratio distributions:
the t-distribution arises from a Gaussian random variable divided by an independent chi-distributed random variable,
while the F-distribution originates from the ratio of two independent chi-squared distributed random variables.
More general ratio distributions have been considered in the literature.
Often the ratio distributions are heavy-tailed, and it may be difficult to work with such distributions and develop an associated statistical test.
A method based on the median has been suggested as a "work-around".
Algebra of random variables
The ratio is one type of algebra for random variables:
Related to the ratio distribution are the product distribution, sum distribution and difference distribution. More generally, one may talk of combinations of sums, differences, products and ratios.
Many of these distributions are described in Melvin D. Springer's book from 1979 The Algebra of Random Variables.
The algebraic rules known with ordinary numbers do not apply for the algebra of random variables.
For example, if a product is C = AB and a ratio is D=C/A it does not necessarily mean that the distributions of D and B are the same.
Indeed, a peculiar effect is seen for the Cauchy distribution: The product and the ratio of two independent Cauchy distributions (with the same scale parameter and the location parameter set to zero) will give the same distribution.

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