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
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 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.
From Mellin transform theory, for distributions existing only on the positive half-line , we have the product identity provided are independent. For the case of a ratio of samples like , in order to make use of this identity it is necessary to use moments of the inverse distribution. Set such that .
Thus, if the moments of can be determined separately, then the moments of can be found. The moments of are determined from the inverse pdf of , often a tractable exercise. At simplest, .
To illustrate, let be sampled from a standard Gamma distribution moment is .
is sampled from an inverse Gamma distribution with parameter and has pdf . The moments of this pdf are .
Multiplying the corresponding moments gives
Independently, it is known that the ratio of the two Gamma samples follows the Beta Prime distribution: whose moments are
Substituting we have
which is consistent with the product of moments above.
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 removed all approximation from Geary's earlier result but his algorithm, as published, is not quite computer-ready due to the Gaussian integral in the final result (eqns 23-24) possibly going backward along the axis which needs to be trapped out. 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:
The above expression becomes even more complicated if the variables X and Y are correlated. In the case that and we have the standard Cauchy distribution. This is most easily derived by a change of variable. Since is uniformly distributed on for the bivariate Normal distribution then in the right hand semicircle we have . Defining we have . Finally set to get and by circular symmetry, .
If , or the more general Cauchy distribution is obtained
Geary showed how the correlated ratio could be transformed into a near-Gaussian form and developed an approximation for dependent on the probability of negative denominator values being vanishingly small. Fieller's later correlated ratio analysis is exact but cumbersome and incompatible with modern math packages without manual intervention to ensure the Normal integral always is defined in a positive direction. The latter problem can also be identified in some of Marsaglia's equations. Hinkley's correlated results are exact but it is shown below that the correlated ratio condition can be transformed simply into an uncorrelated one so only the simplified Hinkley equations above are required, not the full correlated ratio version.
Let the ratio be in which are zero-mean correlated normal variables with variances and have means . We can in general write such that become uncorrelated and has standard deviation . The ratio is invariant and retains the same pdf.
The term in the numerator is made separable by expanding
in which .
Finally, to be explicit, the pdf of the ratio for correlated variables is found by inputting the modified parameters and into the Hinkley equation above which returns the pdf for the correlated ratio with an offset on . In retrospect this transformation will be recognized as being the same as that used by Geary as a partial result in his eqn viii but which is not well-explained and shows that part of Geary's transformation is not dependent on the positivity of Y
The figures below show an example of a positively correlated ratio with in which the shaded areas represent the increment of area selected by given ratio which accumulates probability from the distribution. The theoretical distribution below, derived from the equations under discussion combined with Hinkley's equations, is highly consistent with a simulation result using 5,000 samples. In the top figure it is easily understood that for a ratio the line almost bypasses the distribution mass altogether and this coincides with a near-zero region in the theoretical pdf. Conversely as reduces toward zero the line collects a higher probability.
Contours of the bivariate Gaussian distribution (not to scale)
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 
This distribution does not depend on and 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 :
In the ratios above, Gamma samples, U, V may have differing sample sizes but must be drawn from the same distribution with equal scaling .
In situations where U and V are differently scaled, a variables transformation allows the modified random ratio pdf to be determined.
Let where arbitrary
and, from above, .
Rescale V arbitrarily, defining
We have and substitution into Y gives Transforming X to Y gives Noting we finally have
Thus, if and then is distributed as with
The distribution of Y is limited here to the interval [0,1]. It can be generalized by scaling such that if then
is then a sample from
Though not ratio distributions of two variables, the following identities are useful:
thus, from above,
If X and Y are exponential random variables with mean μ, then X-Y is a double exponential random variable with mean 0 and scale μ.
^ abPham-Gia, T.; Turkkan, N.; Marchand, E. (2006). "Density of the Ratio of Two Normal Random Variables and Applications". Communications in Statistics - Theory and Methods. Taylor & Francis. 35 (9): 1569–1591. doi:10.1080/03610920600683689.
^ abcKermond, John (2010). "An Introduction to the Algebra of Random Variables". Mathematical Association of Victoria 47th Annual Conference Proceedings - New Curriculum. New Opportunities. The Mathematical Association of Victoria: 1–16. ISBN978-1-876949-50-1.
^"SLAPPF". Statistical Engineering Division, National Institute of Science and Technology. Retrieved 2009-07-02.
^ B. Raja Rao, M. L. Garg. "A note on the generalized (positive) Cauchy distribution."
"Canadian Mathematical Bulletin." 12(1969), 865-868 Published:1969-01-01
^Katz D. et al.(1978) Obtaining confidence intervals for the risk ratio in cohort studies. Biometrics 34:469–474