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
Contents
- Algebra of random variables
- Derivation
- Gaussian ratio distribution
- A transformation to Gaussianity
- Uniform ratio distribution
- Cauchy ratio distribution
- Ratio of standard normal to standard uniform
- Other ratio distributions
- Ratio distributions in multivariate analysis
- References
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. 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
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,
where
Derivation
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
This is not always straightforward.
The Mellin transform has also been suggested for derivation of ratio distributions.
Gaussian ratio distribution
When X and Y are independent and have a Gaussian distribution with zero mean the form of their ratio distribution is fairly simple: It is a Cauchy distribution. However, when the two distributions have non-zero means then the form for the distribution of the ratio is much more complicated. In 1932 Fieller found a form for this distribution; here it is given in the more succinct form presented by David Hinkley. In the absence of correlation (cor(X,Y) = 0), the probability density function of the two normal variable X = N(μX, σX2) and Y = N(μY, σY2) ratio Z = X/Y is given by the following expression:
where
And
The above expression becomes even more complicated if the variables X and Y are correlated. It can also be shown that p(z) is a standard Cauchy distribution if μX = μY = 0, and σX = σY = 1. In such case b(z) = 0, and
If
where ρ is the correlation coefficient between X and Y and
The complex distribution has also been expressed with Kummer's confluent hypergeometric function or the Hermite function.
A transformation to Gaussianity
A transformation has been suggested so that, under certain assumptions, the transformed variable T would approximately have a standard Gaussian distribution:
The transformation has been called the Geary–Hinkley transformation, and the approximation is good if Y is unlikely to assume negative values.
Uniform ratio distribution
With two independent random variables following a uniform distribution, e.g.,
the ratio distribution becomes
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
This distribution does not depend on
More generally, if two independent random variables X and Y each follow a Cauchy distribution with median equal to zero and shape factor
1. The ratio distribution for the random variable
2. The product distribution for the random variable
The result for the ratio distribution can be obtained from the product distribution by replacing
Ratio of standard normal to standard uniform
If X has a standard normal distribution and Y has a standard uniform distribution, then Z = X / Y has a distribution known as the slash distribution, with probability density function
where φ(z) is the probability density function of the standard normal distribution.
Other ratio distributions
Let X be a normal(0,1) distribution, Y and Z be a chi square distributions with m and n degrees of freedom respectively. Then
where tm is Student's t distribution,
Ratio distributions in multivariate analysis
Ratio distributions also appear in multivariate analysis. If the random matrices X and Y follow a Wishart distribution then the ratio of the determinants
is proportional to the product of independent F random variables. In the case where X and Y are from independent standardized Wishart distributions then the ratio
has a Wilks' lambda distribution.