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A ratio distribution (or 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 Gaussian (normal) 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.〔George Marsaglia (April 1964). ''(Ratios of Normal Variables and Ratios of Sums of Uniform Variables )''. Defense Technical Information Center.〕 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". ==Algebra of random variables== (詳細は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, and each constructed from two Gaussian distributions and then : where . The first term is the ratio of two Cauchy distributions while the last term is the product of two such distributions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ratio distribution」の詳細全文を読む スポンサード リンク
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