Ratio distribution
More generally, one may talk of combinations of sums, differences, products and ratios.
By way of example take the classical problem of the ratio of two standard Gaussian samples.
Secondly, integrating the horizontal strips upward over all y yields the volume of probability above the line Finally, differentiate
R yields the pdf of R From Mellin transform theory, for distributions existing only on the positive half-line
In the case of ratios, we have which, in terms of probability distributions, is equivalent to Note that
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.
, the probability density function of the ratio Z = X/Y of two normal variables X = N(μX, σX2) and Y = N(μY, σY2) is given exactly by the following expression, derived in several sources:[6] where The above expression becomes more complicated when the variables X and Y are correlated.
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
Fieller's later correlated ratio analysis is exact but care is needed when combining modern math packages with verbal conditions in the older literature.
has standard deviation The ratio: is invariant under this transformation and retains the same pdf.
into the Hinkley equation above which returns the pdf for the correlated ratio with a constant offset
in which the shaded wedges represent the increment of area selected by given ratio
the wedge has almost bypassed the main distribution mass altogether and this explains the local minimum in the theoretical pdf
moves either toward or away from one the wedge spans more of the central mass, accumulating a higher probability.
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.
The graph shows the pdf of the ratio of two complex normal variables with a correlation coefficient of
[note 1] This is important for many applications requiring the ratio of random variables that must be positive, where joint distribution of
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.
: More generally, if two independent random variables X and Y each follow a Cauchy distribution with median equal to zero and shape factor
, 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.
follows and has cdf The generalized gamma distribution is which includes the regular gamma, chi, chi-squared, exponential, Rayleigh, Nakagami and Weibull distributions involving fractional powers.
In situations where U and V are differently scaled, a variables transformation allows the modified random ratio pdf to be determined.
A number of papers compare the robustness of different approximations for the binomial ratio.
[citation needed] 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.
, the generic pdf of a left truncated Poisson distribution is which sums to unity.
Absent any closed form solutions, the following approximate reversion for truncated
