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.[1][2][3][4][5][6][7][8][9]

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".[10]

Algebra of random variables edit

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.[8]

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.[8] This becomes evident when regarding the Cauchy distribution as itself a ratio distribution of two Gaussian distributions of zero means: 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.

Derivation edit

A way of deriving the ratio distribution of   from the joint distribution of the two other random variables X , Y , with joint pdf  , is by integration of the following form[3]


If the two variables are independent then   and this becomes


This may not be straightforward. By way of example take the classical problem of the ratio of two standard Gaussian samples. The joint pdf is


Defining   we have


Using the known definite integral   we get


which is the Cauchy distribution, or Student's t distribution with n = 1

The Mellin transform has also been suggested for derivation of ratio distributions.[8]

In the case of positive independent variables, proceed as follows. The diagram shows a separable bivariate distribution   which has support in the positive quadrant   and we wish to find the pdf of the ratio  . The hatched volume above the line   represents the cumulative distribution of the function   multiplied with the logical function  . The density is first integrated in horizontal strips; the horizontal strip at height y extends from x = 0 to x = Ry and has incremental probability  .
Secondly, integrating the horizontal strips upward over all y yields the volume of probability above the line


Finally, differentiate   with respect to   to get the pdf  .


Move the differentiation inside the integral:


and since




As an example, find the pdf of the ratio R when

Evaluating the cumulative distribution of a ratio

We have




Differentiation wrt. R yields the pdf of R


Moments of random ratios edit

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   and   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

  whose  -th 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.

Means and variances of random ratios edit

In the Product distribution section, and derived from Mellin transform theory (see section above), it is found that the mean of a product of independent variables is equal to the product of their means. In the case of ratios, we have


which, in terms of probability distributions, is equivalent to


Note that   i.e.,  

The variance of a ratio of independent variables is


Normal ratio distributions edit

Uncorrelated central normal ratio edit

When X and Y are independent and have a Gaussian distribution with zero mean, the form of their ratio distribution is a Cauchy distribution. This can be derived by setting   then showing that   has circular symmetry. For a bivariate uncorrelated Gaussian distribution we have


If   is a function only of r then   is uniformly distributed on   with density   so the problem reduces to finding the probability distribution of Z under the mapping


We have, by conservation of probability


and since  


and setting   we get


There is a spurious factor of 2 here. Actually, two values of   spaced by   map onto the same value of z, the density is doubled, and the final result is


When either of the two Normal distributions is non-central then the result for the distribution of the ratio is much more complicated and is given below in the succinct form presented by David Hinkley.[6] The trigonometric method for a ratio does however extend to radial distributions like bivariate normals or a bivariate Student t in which the density depends only on radius  . It does not extend to the ratio of two independent Student t distributions which give the Cauchy ratio shown in a section below for one degree of freedom.

Uncorrelated noncentral normal ratio edit

In the absence of correlation  , the probability density function of the two normal variables X = N(μX, σX2) and Y = N(μY, σY2) ratio Z = X/Y is given exactly by the following expression, derived in several sources:[6]




and   is the normal cumulative distribution function:

  • Under several assumptions (usually fulfilled in practical applications), it is possible to derive a highly accurate solid approximation to the PDF. Main benefits are reduced formulae complexity, closed-form CDF, simple defined median, well defined error management, etc... For the sake of simplicity let's introduce parameters:  ,   and  . Then so called solid approximation   to the uncorrelated noncentral normal ratio PDF is expressed by equation [11]
  • Under certain conditions, a normal approximation is possible, with variance:[12]

Correlated central normal ratio edit

The above expression becomes more complicated when the variables X and Y are correlated. If   but   and   the more general Cauchy distribution is obtained


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.[9]

Correlated noncentral normal ratio edit

This was shown in Springer 1979 problem 4.28.

A transformation to the log domain was suggested by Katz(1978) (see binomial section below). Let the ratio be


Take logs to get


Since   then asymptotically


Alternatively, Geary (1930) suggested that


has approximately a standard Gaussian distribution:[1] This transformation has been called the Geary–Hinkley transformation;[7] the approximation is good if Y is unlikely to assume negative values, basically  .

Exact correlated noncentral normal ratio edit

This is developed by Dale (Springer 1979 problem 4.28) and Hinkley 1969. 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 care is needed when combining modern math packages with verbal conditions in the older literature. Pham-Ghia has exhaustively discussed these methods. Hinkley's correlated results are exact but it is shown below that the correlated ratio condition can also be transformed 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   Write   such that   become uncorrelated and   has standard deviation


The ratio:


is invariant under this transformation and retains the same pdf. The   term in the numerator appears to be made separable by expanding:


to get


in which   and z has now become a ratio of uncorrelated non-central normal samples with an invariant z-offset (this is not formally proven, though appears to have been used by Geary),

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 a constant offset   on  .

Contours of the correlated bivariate Gaussian distribution (not to scale) giving ratio x/y
pdf of the Gaussian ratio z and a simulation (points) for

The figures above show an example of a positively correlated ratio with   in which the shaded wedges represent the increment of area selected by given ratio   which accumulates probability where they overlap the distribution. The theoretical distribution, 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 clear that for a ratio   the wedge has almost bypassed the main distribution mass altogether and this explains the local minimum in the theoretical pdf  . Conversely as   moves either toward or away from one the wedge spans more of the central mass, accumulating a higher probability.

Complex normal ratio edit

The ratio of correlated zero-mean circularly symmetric complex normal distributed variables was determined by Baxley et al.[13] and has since been extended to the nonzero-mean and nonsymmetric case.[14] In the correlated zero-mean case, the joint distribution of x, y is




  is an Hermitian transpose and


The PDF of   is found to be


In the usual event that   we get


Further closed-form results for the CDF are also given.

The ratio distribution of correlated complex variables, rho = 0.7 exp(i pi/4).

The graph shows the pdf of the ratio of two complex normal variables with a correlation coefficient of  . The pdf peak occurs at roughly the complex conjugate of a scaled down  .

Ratio of log-normal edit

The ratio of independent or correlated log-normals is log-normal. This follows, because if   and   are log-normally distributed, then   and   are normally distributed. If they are independent or their logarithms follow a bivarate normal distribution, then the logarithm of their ratio is the difference of independent or correlated normally distributed random variables, which is normally distributed.[note 1]

This is important for many applications requiring the ratio of random variables that must be positive, where joint distribution of   and   is adequately approximated by a log-normal. This is a common result of the multiplicative central limit theorem, also known as Gibrat's law, when   is the result of an accumulation of many small percentage changes and must be positive and approximately log-normally distributed.[15]

Uniform ratio distribution edit

With two independent random variables following a uniform distribution, e.g.,


the ratio distribution becomes


Cauchy ratio distribution edit

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[16]


This distribution does not depend on   and the result stated by Springer[8] (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  :


More generally, if two independent random variables X and Y each follow a Cauchy distribution with median equal to zero and shape factor   and   respectively, then:

  1. The ratio distribution for the random variable   is[16]
  2. The product distribution for the random variable   is[16]

The result for the ratio distribution can be obtained from the product distribution by replacing   with  

Ratio of standard normal to standard uniform edit

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.[17]

Chi-squared, Gamma, Beta distributions edit

Let G be a normal(0,1) distribution, Y and Z be chi-squared distributions with m and n degrees of freedom respectively, all independent, with  . Then

  the Student's t distribution
  i.e. Fisher's F-test distribution
  the beta distribution
  the standard beta prime distribution

If  , a noncentral chi-squared distribution, and   and   is independent of   then

 , a noncentral F-distribution.

  defines  , Fisher's F density distribution, the PDF of the ratio of two Chi-squares with m, n degrees of freedom.

The CDF of the Fisher density, found in F-tables is defined in the beta prime distribution article. If we enter an F-test table with m = 3, n = 4 and 5% probability in the right tail, the critical value is found to be 6.59. This coincides with the integral


For gamma distributions U and V with arbitrary shape parameters α1 and α2 and their scale parameters both set to unity, that is,  , where  , then


If  , then  . Note that here θ is a scale parameter, rather than a rate parameter.

If  , then by rescaling the   parameter to unity we have




in which   represents the generalised beta prime distribution.

In the foregoing it is apparent that if   then  . More explicitly, since


if   then




Rayleigh Distributions edit

If X, Y are independent samples from the Rayleigh distribution  , the ratio Z = X/Y follows the distribution[18]


and has cdf


The Rayleigh distribution has scaling as its only parameter. The distribution of   follows


and has cdf


Fractional gamma distributions (including chi, chi-squared, exponential, Rayleigh and Weibull) edit

The generalized gamma distribution is


which includes the regular gamma, chi, chi-squared, exponential, Rayleigh, Nakagami and Weibull distributions involving fractional powers. Note that here a is a scale parameter, rather than a rate parameter; d is a shape parameter.


Modelling a mixture of different scaling factors edit

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  

Reciprocals of samples from beta distributions edit

Though not ratio distributions of two variables, the following identities for one variable are useful:

If   then  
If   then  

combining the latter two equations yields

If   then  .
If   then  


 , the distribution of the reciprocals of   samples.

If   then   and


Further results can be found in the Inverse distribution article.

  • If   are independent exponential random variables with mean μ, then X − Y is a double exponential random variable with mean 0 and scale μ.

Binomial distribution edit

This result was derived by Katz et al.[20]

Suppose   and   and  ,   are independent. Let  .

Then   is approximately normally distributed with mean   and variance  .

The binomial ratio distribution is of significance in clinical trials: if the distribution of T is known as above, the probability of a given ratio arising purely by chance can be estimated, i.e. a false positive trial. A number of papers compare the robustness of different approximations for the binomial ratio.[citation needed]

Poisson and truncated Poisson distributions edit

In the ratio of Poisson variables R = X/Y there is a problem that Y is zero with finite probability so R is undefined. To counter this, consider the truncated, or censored, ratio R' = X/Y' where zero sample of Y are discounted. Moreover, in many medical-type surveys, there are systematic problems with the reliability of the zero samples of both X and Y and it may be good practice to ignore the zero samples anyway.

The probability of a null Poisson sample being  , the generic pdf of a left truncated Poisson distribution is


which sums to unity. Following Cohen,[21] for n independent trials, the multidimensional truncated pdf is


and the log likelihood becomes


On differentiation we get


and setting to zero gives the maximum likelihood estimate  


Note that as   then   so the truncated maximum likelihood   estimate, though correct for both truncated and untruncated distributions, gives a truncated mean   value which is highly biassed relative to the untruncated one. Nevertheless it appears that   is a sufficient statistic for   since   depends on the data only through the sample mean   in the previous equation which is consistent with the methodology of the conventional Poisson distribution.

Absent any closed form solutions, the following approximate reversion for truncated   is valid over the whole range  .


which compares with the non-truncated version which is simply  . Taking the ratio   is a valid operation even though   may use a non-truncated model while   has a left-truncated one.

The asymptotic large-  (and Cramér–Rao bound) is


in which substituting L gives


Then substituting   from the equation above, we get Cohen's variance estimate


The variance of the point estimate of the mean  , on the basis of n trials, decreases asymptotically to zero as n increases to infinity. For small   it diverges from the truncated pdf variance in Springael[22] for example, who quotes a variance of


for n samples in the left-truncated pdf shown at the top of this section. Cohen showed that the variance of the estimate relative to the variance of the pdf,  , ranges from 1 for large   (100% efficient) up to 2 as   approaches zero (50% efficient).

These mean and variance parameter estimates, together with parallel estimates for X, can be applied to Normal or Binomial approximations for the Poisson ratio. Samples from trials may not be a good fit for the Poisson process; a further discussion of Poisson truncation is by Dietz and Bohning[23] and there is a Zero-truncated Poisson distribution Wikipedia entry.

Double Lomax distribution edit

This distribution is the ratio of two Laplace distributions.[24] Let X and Y be standard Laplace identically distributed random variables and let z = X / Y. Then the probability distribution of z is


Let the mean of the X and Y be a. Then the standard double Lomax distribution is symmetric around a.

This distribution has an infinite mean and variance.

If Z has a standard double Lomax distribution, then 1/Z also has a standard double Lomax distribution.

The standard Lomax distribution is unimodal and has heavier tails than the Laplace distribution.

For 0 < a < 1, the a-th moment exists.


where Γ is the gamma function.

Ratio distributions in multivariate analysis edit

Ratio distributions also appear in multivariate analysis.[25] 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.

Ratios of Quadratic Forms involving Wishart Matrices edit

In relation to Wishart matrix distributions if   is a sample Wishart matrix and vector   is arbitrary, but statistically independent, corollary 3.2.9 of Muirhead[26] states


The discrepancy of one in the sample numbers arises from estimation of the sample mean when forming the sample covariance, a consequence of Cochran's theorem. Similarly


which is Theorem 3.2.12 of Muirhead [26]

See also edit

Notes edit

  1. ^ Note, however, that   and   can be individually log-normally distributed without having a bivariate log-normal distribution. As of 2022-06-08 the Wikipedia article on "Copula (probability theory)" includes a density and contour plot of two Normal marginals joint with a Gumbel copula, where the joint distribution is not bivariate normal.

References edit

  1. ^ a b Geary, R. C. (1930). "The Frequency Distribution of the Quotient of Two Normal Variates". Journal of the Royal Statistical Society. 93 (3): 442–446. doi:10.2307/2342070. JSTOR 2342070.
  2. ^ Fieller, E. C. (November 1932). "The Distribution of the Index in a Normal Bivariate Population". Biometrika. 24 (3/4): 428–440. doi:10.2307/2331976. JSTOR 2331976.
  3. ^ a b Curtiss, J. H. (December 1941). "On the Distribution of the Quotient of Two Chance Variables". The Annals of Mathematical Statistics. 12 (4): 409–421. doi:10.1214/aoms/1177731679. JSTOR 2235953.
  4. ^ George Marsaglia (April 1964). Ratios of Normal Variables and Ratios of Sums of Uniform Variables. Defense Technical Information Center.
  5. ^ Marsaglia, George (March 1965). "Ratios of Normal Variables and Ratios of Sums of Uniform Variables". Journal of the American Statistical Association. 60 (309): 193–204. doi:10.2307/2283145. JSTOR 2283145. Archived from the original on September 23, 2017.
  6. ^ a b c Hinkley, D. V. (December 1969). "On the Ratio of Two Correlated Normal Random Variables". Biometrika. 56 (3): 635–639. doi:10.2307/2334671. JSTOR 2334671.
  7. ^ a b Hayya, Jack; Armstrong, Donald; Gressis, Nicolas (July 1975). "A Note on the Ratio of Two Normally Distributed Variables". Management Science. 21 (11): 1338–1341. doi:10.1287/mnsc.21.11.1338. JSTOR 2629897.
  8. ^ a b c d e f Springer, Melvin Dale (1979). The Algebra of Random Variables. Wiley. ISBN 0-471-01406-0.
  9. ^ a b Pham-Gia, T.; Turkkan, N.; Marchand, E. (2006). "Density of the Ratio of Two Normal Random Variables and Applications". Communications in Statistics – Theory and Methods. 35 (9). Taylor & Francis: 1569–1591. doi:10.1080/03610920600683689. S2CID 120891296.
  10. ^ Brody, James P.; Williams, Brian A.; Wold, Barbara J.; Quake, Stephen R. (October 2002). "Significance and statistical errors in the analysis of DNA microarray data" (PDF). Proc Natl Acad Sci U S A. 99 (20): 12975–12978. Bibcode:2002PNAS...9912975B. doi:10.1073/pnas.162468199. PMC 130571. PMID 12235357.
  11. ^ Šimon, Ján; Ftorek, Branislav (2022-09-15). "Basic Statistical Properties of the Knot Efficiency". Symmetry. 14 (9). MDPI: 1926. doi:10.3390/sym14091926. ISSN 2073-8994.
  12. ^ Díaz-Francés, Eloísa; Rubio, Francisco J. (2012-01-24). "On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables". Statistical Papers. 54 (2). Springer Science and Business Media LLC: 309–323. doi:10.1007/s00362-012-0429-2. ISSN 0932-5026. S2CID 122038290.
  13. ^ Baxley, R T; Waldenhorst, B T; Acosta-Marum, G (2010). "Complex Gaussian Ratio Distribution with Applications for Error Rate Calculation in Fading Channels with Imperfect CSI". 2010 IEEE Global Telecommunications Conference GLOBECOM 2010. pp. 1–5. doi:10.1109/GLOCOM.2010.5683407. ISBN 978-1-4244-5636-9. S2CID 14100052.
  14. ^ Sourisseau, M.; Wu, H.-T.; Zhou, Z. (October 2022). "Asymptotic analysis of synchrosqueezing transform—toward statistical inference with nonlinear-type time-frequency analysis". Annals of Statistics. 50 (5): 2694–2712. arXiv:1904.09534. doi:10.1214/22-AOS2203.
  15. ^ Of course, any invocation of a central limit theorem assumes suitable, commonly met regularity conditions, e.g., finite variance.
  16. ^ a b c Kermond, 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. ISBN 978-1-876949-50-1.
  17. ^ "SLAPPF". Statistical Engineering Division, National Institute of Science and Technology. Retrieved 2009-07-02.
  18. ^ Hamedani, G. G. (Oct 2013). "Characterizations of Distribution of Ratio of Rayleigh Random Variables". Pakistan Journal of Statistics. 29 (4): 369–376.
  19. ^ Raja Rao, B.; Garg., M. L. (1969). "A note on the generalized (positive) Cauchy distribution". Canadian Mathematical Bulletin. 12 (6): 865–868. doi:10.4153/CMB-1969-114-2.
  20. ^ Katz D. et al.(1978) Obtaining confidence intervals for the risk ratio in cohort studies. Biometrics 34:469–474
  21. ^ Cohen, A Clifford (June 1960). "Estimating the Parameter in a Conditional Poisson Distribution". Biometrics. 60 (2): 203–211. doi:10.2307/2527552. JSTOR 2527552.
  22. ^ Springael, Johan (2006). "On the sum of independent zero-truncated Poisson random variables" (PDF). University of Antwerp, Faculty of Business and Economics.
  23. ^ Dietz, Ekkehart; Bohning, Dankmar (2000). "On Estimation of the Poisson Parameter in Zero-Modified Poisson Models". Computational Statistics & Data Analysis. 34 (4): 441–459. doi:10.1016/S0167-9473(99)00111-5.
  24. ^ Bindu P and Sangita K (2015) Double Lomax distribution and its applications. Statistica LXXV (3) 331–342
  25. ^ Brennan, L E; Reed, I S (January 1982). "An Adaptive Array Signal Processing Algorithm for Communications". IEEE Transactions on Aerospace and Electronic Systems. AES-18 No 1: 124–130. Bibcode:1982ITAES..18..124B. doi:10.1109/TAES.1982.309212. S2CID 45721922.
  26. ^ a b Muirhead, Robb (1982). Aspects of Multivariate Statistical Theory. USA: Wiley. pp. 96, Theorem 3.2.12.

External links edit