Ratio distribution

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 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 is a ratio distribution. The Cauchy distribution is an example of a ratio distribution. The random variable associated with this distribution comes about as the ratio of two normally distributed variables with zero mean. Thus the Cauchy distribution is also called the normal ratio distribution.A number of researchers have considered more general ratio distributions.Two distributions often used in test-statistics, the t-distribution and the F-distribution, are also ratio distributions: The t-distributed random variable is the ratio of a Gaussian random variable divided by an independent chi-distributed random variable (i.e., the square root of a chi-squared distribution), while the F-distributed random variable is the ratio of two independent chi-squared distributed random variables. 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'. 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.This becomes evident when regarding the Cauchy distribution as itself a ratio distribution of two Gaussian distributions: Consider two Cauchy random variables, C 1 {displaystyle C_{1}} and C 2 {displaystyle C_{2}} each constructed from two Gaussian distributions C 1 = G 1 / G 2 {displaystyle C_{1}=G_{1}/G_{2}} and C 2 = G 3 / G 4 {displaystyle C_{2}=G_{3}/G_{4}} then where C 3 = G 4 / G 3 {displaystyle C_{3}=G_{4}/G_{3}} . The first term is the ratio of two Cauchy distributions while the last term is the product of two such distributions. A way of deriving the ratio distribution of Z from the joint distribution of the two other random variables, X and Y, is by integration of the following form