Ratio distribution#Normal

{{main|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 of zero means: Consider two Cauchy random variables, $C\_1$ and $C\_2$ each constructed from two Gaussian distributions $C\_1=G\_1/G\_2$ and $C\_2\; =\; G\_3/G\_4$ then

: $\backslash frac\{C\_1\}\{C\_2\}\; =\; \backslash frac\{\{G\_1\}/\{G\_2\}\}\{\{G\_3\}/\{G\_4\}\}\; =\; \backslash frac\{G\_1\; G\_4\}\{G\_2\; G\_3\}\; =\; \backslash frac\{G\_1\}\{G\_2\}\; \backslash times\; \backslash frac\{G\_4\}\{G\_3\}\; =\; C\_1\; \backslash times\; C\_3,$

where $C\_3\; =\; G\_4/G\_3$. The first term is the ratio of two Cauchy distributions while the last term is the product of two such distributions.

A way of deriving the ratio distribution of $Z\; =\; X/Y$ from the joint distribution of the two other random variables X , Y , with joint pdf $p\_\{X,Y\}(x,y)$, is by integration of the following form

: $p\_Z(z)\; =\; \backslash int^\{+\backslash infty\}\_\{-\backslash infty\}\; |y|\backslash ,\; p\_\{X,Y\}(zy,\; y)\; \backslash ,\; dy.$

If the two variables are independent then $p\_\{XY\}(x,y)\; =\; p\_X(x)\; p\_Y(y)$ and this becomes

: $p\_Z(z)\; =\; \backslash int^\{+\backslash infty\}\_\{-\backslash infty\}\; |y|\backslash ,\; p\_X(zy)\; p\_Y(y)\; \backslash ,\; dy.$

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

:$p\_\{X,Y\}(x,y)\; =\; \backslash frac\; \{1\}\{2\; \backslash pi\; \}\backslash exp\backslash left(-\backslash frac\{x^2\}\{2\}\; \backslash right)\; \backslash exp\; \backslash left(-\backslash frac\{y^2\}\{2\}\; \backslash right)$

Defining $Z\; =\; X/Y$ we have

:$\backslash begin\{align\}$

p_Z(z) &= \frac {1}{2 \pi }\int_{-\infty}^{\infty} \, |y| \, \exp\left(-\frac{\left(zy\right)^2}{2} \right) \, \exp\left(-\frac{ y^2}{2} \right) \, dy \\

&= \frac {1}{2 \pi } \int_{-\infty}^{\infty} \,|y| \, \exp\left(-\frac{y^2 \left(z^2 + 1\right)}{2} \right) \, dy

\end{align}

Using the known definite integral $\backslash int\_0^\{\backslash infty\}\; \backslash ,\; x\; \backslash ,\; \backslash exp\backslash left(-cx^2\; \backslash right)\; \backslash ,\; dx\; =\; \backslash frac\; \{1\}\{2c\}$ we get

:$p\_Z(z)\; =\; \backslash frac\; \{1\}\{\; \backslash pi\; (z^2\; +\; 1)\}$

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.

In the case of positive independent variables, proceed as follows. The diagram shows a separable bivariate distribution $f\_\{x,y\}(x,y)=f\_x(x)f\_y(y)$ which has support in the positive quadrant $x,y\; >\; 0$ and we wish to find the pdf of the ratio $R=\; X/Y$. The hatched volume above the line $y\; =\; x/\; R$ represents the cumulative distribution of the function $f\_\{x,y\}(x,y)$ multiplied with the logical function $X/Y\; \backslash le\; R$. 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 $f\_y(y)dy\; \backslash int\_0^\{Ry\}\; f\_x(x)\; \backslash ,dx$.

Secondly, integrating the horizontal strips upward over all y yields the volume of probability above the line

:$F\_R(R)\; =\; \backslash int\_0^\backslash infty\; f\_y(y)\; \backslash left(\backslash int\_0^\{Ry\}\; f\_x(x)dx\; \backslash right)\; dy$

Finally, differentiate $F\_R(R)$ with respect to $R$ to get the pdf $f\_R(R)$.

:$f\_R(R)\; =\; \backslash frac\{d\}\{dR\}\; \backslash left[\; \backslash int\_0^\backslash infty\; f\_y(y)\; \backslash left(\backslash int\_0^\{Ry\}\; f\_x(x)dx\; \backslash right)\; dy\; \backslash right]$

Move the differentiation inside the integral:

:$f\_R(R)\; =\; \backslash int\_0^\backslash infty\; f\_y(y)\; \backslash left(\backslash frac\{d\}\{dR\}\; \backslash int\_0^\{Ry\}\; f\_x(x)dx\; \backslash right)\; dy$

and since

:$\backslash frac\{d\}\{dR\}\; \backslash int\_0^\{Ry\}\; f\_x(x)dx\; =\; yf\_x(Ry)$

then

:$f\_R(R)\; =\; \backslash int\_0^\backslash infty\; f\_y(y)\; \backslash ;\; f\_x(Ry)\; \backslash ;\; y\; \backslash ;\; dy$

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

: $f\_x(x)\; =\; \backslash alpha\; e^\{-\backslash alpha\; x\},\; \backslash ;\backslash ;\backslash ;\backslash ;\; f\_y(y)\; =\; \backslash beta\; e^\{-\backslash beta\; y\},\; \backslash ;\backslash ;\backslash ;\; x,y\; \backslash ge\; 0$

File:RatioDistribution.jpg

We have

:$\backslash int\_0^\{Ry\}\; f\_x(x)dx\; =\; -\; e^\{-\backslash alpha\; x\}\; \backslash vert\_0^\{Ry\}\; =\; 1-\; e^\{-\backslash alpha\; Ry\}$

thus

:

&=\int_0^\infty \beta e^{-\beta y} \left( 1- e^{-\alpha Ry} \right) dy \\

& = 1 - \frac{\alpha R}{\beta + \alpha R} \\

& = \frac{ R}{\tfrac{\beta}{\alpha} + R}

\end{align}

Differentiation wrt. R yields the pdf of R

:$f\_R(R)\; =\backslash frac\{d\}\{dR\}\; \backslash left(\; \backslash frac\{\; R\}\{\backslash tfrac\{\backslash beta\}\{\backslash alpha\}\; +\; R\}\; \backslash right)\; =\; \backslash frac\{\backslash tfrac\{\backslash beta\}\{\backslash alpha\}\}\; \{\backslash left(\; \backslash tfrac\{\backslash beta\}\{\backslash alpha\}\; +\; R\; \backslash right)^2\; \}$

From Mellin transform theory, for distributions existing only on the positive half-line $x\; \backslash ge\; 0$, we have the product identity $\backslash operatorname\{E\}[(UV)^p\; ]\; =\; \backslash operatorname\{E\}[U^p\; ]\; \backslash ;\backslash ;\; \backslash operatorname\{E\}[V^p\; ]$ provided $U,\; \backslash ;\; V$ are independent. For the case of a ratio of samples like $\backslash operatorname\{E\}[(X/Y)^p]$, in order to make use of this identity it is necessary to use moments of the inverse distribution. Set $1/Y\; =\; Z$ such that $\backslash operatorname\{E\}[(XZ)^p\; ]\; =\; \backslash operatorname\{E\}[X^p\; ]\; \backslash ;\; \backslash operatorname\{E\}[Y^\{-p\}\; ]$.

Thus, if the moments of $X^p$ and $Y^\{-p\}$ can be determined separately, then the moments of $X/Y$ can be found. The moments of $Y^\{-p\}$ are determined from the inverse pdf of $Y$ , often a tractable exercise. At simplest, $\backslash operatorname\{E\}[\; Y^\{-p\}\; ]\; =\backslash int\_0^\backslash infty\; y^\{-p\}\; f\_y(y)\backslash ,dy$.

To illustrate, let $X$ be sampled from a standard Gamma distribution

: $x^\{\backslash alpha\; -\; 1\}e^\{-x\}/\backslash Gamma(\backslash alpha)$ whose $p$-th moment is $\backslash Gamma(\backslash alpha\; +\; p)\; /\; \backslash Gamma(\backslash alpha)$.

$Z\; =\; Y^\{-1\}$ is sampled from an inverse Gamma distribution with parameter $\backslash beta$ and has pdf $\backslash ;\; \backslash Gamma^\{-1\}(\backslash beta)\; z^\{-(1+\backslash beta)\}\; e^\{-1/z\}$. The moments of this pdf are

:

Multiplying the corresponding moments gives

:

Independently, it is known that the ratio of the two Gamma samples $R\; =\; X/Y$ follows the Beta Prime distribution:

:$f\_\{\backslash beta\text{'}\}(r,\; \backslash alpha,\; \backslash beta)\; =\; B(\backslash alpha,\; \backslash beta)^\{-1\}\; r^\{\backslash alpha-1\}\; (1+r)^\{-(\backslash alpha\; +\; \backslash beta)\}$ whose moments are $\backslash operatorname\{E\}[R^p]=\; \backslash frac\; \{\; \backslash Beta(\backslash alpha\; +\; p,\backslash beta-p)\}\{\; \backslash Beta(\backslash alpha,\; \backslash beta)\; \}$

Substituting $\backslash Beta(\backslash alpha,\; \backslash beta)\; =\backslash frac\; \{\; \backslash Gamma(\backslash alpha)\backslash Gamma(\backslash beta)\}\{\; \backslash Gamma(\backslash alpha\; +\backslash beta)\; \}$ we have

$\backslash operatorname\{E\}[R^p]\; =\; \backslash frac\; \{\; \backslash Gamma(\backslash alpha\; +\; p)\backslash Gamma(\backslash beta\; -\; p)\}\; \{\; \backslash Gamma(\backslash alpha\; +\backslash beta)\; \}\; \backslash Bigg/$

\frac { \Gamma(\alpha)\Gamma(\beta)} { \Gamma(\alpha +\beta) } =

\frac { \Gamma(\alpha +p)\Gamma(\beta - p)} { \Gamma(\alpha) \Gamma(\beta) }

which is consistent with the product of moments above.

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

:$\backslash operatorname\{E\}(X/Y)\; =\; \backslash operatorname\{E\}(X)\backslash operatorname\{E\}(1/Y)$

which, in terms of probability distributions, is equivalent to

: $\backslash operatorname\{E\}(X/Y)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; x\; f\_x(x)\; \backslash ,\; dx\; \backslash times\; \backslash int\_\{-\backslash infty\}^\backslash infty\; y^\{-1\}\; f\_y(y)\; \backslash ,\; dy$

Note that $\backslash operatorname\{E\}(1/Y)\; \backslash neq\; \backslash frac\{1\}\{\backslash operatorname\{E\}(Y)\}$ i.e., $$

\int_{-\infty}^\infty y^{-1} f_y(y) \, dy \ne \frac{1}{\int_{-\infty}^\infty y f_y(y) \, dy}

The variance of a ratio of independent variables is

:

\\ & = \operatorname{E}(X^2) \operatorname{E}(1/Y^2) -

\operatorname{E}^2(X) \operatorname{E}^2(1/Y)

\end{align}

{{split section|Normal ratio distributions|reason=Long section about notable concept.|date=March 2021}}

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 $Z\; =\; X/Y\; =\; \backslash tan\; \backslash theta$ then showing that $\backslash theta$ has circular symmetry. For a bivariate uncorrelated Gaussian distribution we have

:

\tfrac{1}{\sqrt {2\pi}} e^{-\frac{1}{2} y^2 }

\\ &= \tfrac{1}{ 2\pi} e^{-\frac{1}{2} (x^2 + y^2 )}

\\ & = \tfrac{1}{ 2\pi} e^{-\frac{1}{2} r^2 } \text{ with } r^2 = x^2 + y^2

\end{align}

If $p(x,y)$ is a function only of r then $\backslash theta$ is uniformly distributed on $[0,\; 2\backslash pi\; ]$ with density $1/2\backslash pi$ so the problem reduces to finding the probability distribution of Z under the mapping

: $Z\; =\; X/Y\; =\; \backslash tan\; \backslash theta$

We have, by conservation of probability

: $p\_z(z)\; |dz|\; =\; p\_\{\backslash theta\}(\backslash theta)|d\backslash theta|$

and since $dz/d\backslash theta\; =\; 1/\; \backslash cos^2\; \backslash theta$

: $p\_z(z)\; =\; \backslash frac\{p\_\{\backslash theta\}(\backslash theta)\}\{\; |dz/d\backslash theta|\; \}\; =\; \backslash tfrac\{1\}\{2\backslash pi\}\{\backslash cos^2\; \backslash theta\; \}$

and setting $\backslash cos^2\; \backslash theta\; =\; \backslash frac\{1\}\{1+\; (\backslash tan\; \backslash theta)^2\}=\; \backslash frac\{1\}\{1+z^2\}$ we get

: $p\_z(z)\; =\; \backslash frac\{1/(2\backslash pi)\}\{1+z^2\; \}$

There is a spurious factor of 2 here. Actually, two values of $\backslash theta$ spaced by $\backslash pi$ 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 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 $r\; =\; \backslash sqrt\{\; x^2\; +\; y^2\; \}$. 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.

In the absence of correlation $(\backslash operatorname\{cor\}(X,Y)=0)$, the probability density function of the ratio Z = X/Y of two normal variables X = N(μ_X, σ_X²) and Y = N(μ_Y, σ_Y²) is given exactly by the following expression, derived in several sources:

: $p\_Z(z)=\; \backslash left(\; \backslash frac\{1\}\{a^2(z)\; \backslash cdot\; 2\; \backslash pi\; \backslash sigma\_x\; \backslash sigma\_y\}\; \backslash exp\; \backslash left(-\; \backslash frac\{c\}\{2\}\; \backslash right)\; \backslash right)\; \backslash left(\backslash sqrt\{2\; \backslash pi\; \}\; \backslash frac\{b(z)\}\{a(z)\}\; \backslash exp\; \backslash left(\backslash frac\{b^2(z)\}\{2a^2(z)\}\; \backslash right)\; \backslash mathrm\{erf\}\; \backslash left(\backslash frac\{b(z)\}\{\backslash sqrt\{2\}\; a(z)\}\; \backslash right)\; +\; 2\; \backslash right)$

where

: $a(z)=\; \backslash sqrt\{\backslash frac\{1\}\{\backslash sigma\_x^2\}\; z^2\; +\; \backslash frac\{1\}\{\backslash sigma\_y^2\}\}$

: $b(z)=\; \backslash frac\{\backslash mu\_x\; \}\{\backslash sigma\_x^2\}\; z\; +\; \backslash frac\{\backslash mu\_y\}\{\backslash sigma\_y^2\}$

: $c\; =\; \backslash frac\{\backslash mu\_x^2\}\{\backslash sigma\_x^2\}\; +\; \backslash frac\{\backslash mu\_y^2\}\{\backslash sigma\_y^2\}$.

• 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 introduce parameters: $p=\backslash frac\{\backslash mu\_x\}\{\backslash sqrt\{2\}\backslash sigma\_x\}$, $q=\backslash frac\{\backslash mu\_y\}\{\backslash sqrt\{2\}\backslash sigma\_y\}$ and $r=\backslash frac\{\backslash mu\_x\}\{\backslash mu\_y\}$. Then so called solid approximation $p\_Z^\backslash dagger(z)$ to the uncorrelated noncentral normal ratio PDF is expressed by equation {{cite journal | last1=Šimon | first1=Ján | last2=Ftorek | first2=Branislav | title=Basic Statistical Properties of the Knot Efficiency | journal=Symmetry | publisher=MDPI | volume=14 | issue=9 | date=2022-09-15 | issn=2073-8994 | doi=10.3390/sym14091926 | pages=1926 | doi-access=free | bibcode=2022Symm...14.1926S }}

: $p\_Z^\backslash dagger(z)=\backslash frac\{1\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash frac\{p\}\{\backslash mathrm\{erf\}[q]\}\; \backslash frac\{1\}\{r\}\; \backslash frac\{1+\backslash frac\{p^2\}\{q^2\}\backslash frac\{z\}\{r\}\}\{\backslash left(1+\backslash frac\{p^2\}\{q^2\}\backslash left[\backslash frac\{z\}\{r\}\backslash right]^2\backslash right)^\backslash frac\{3\}\{2\}\}\; e^\{-\backslash frac\{p^2\backslash left(\backslash frac\{z\}\{r\}-1\; \backslash right)^2\}\{1+\backslash frac\{p^2\}\{q^2\}\backslash left[\backslash frac\{z\}\{r\}\backslash right]^2\}\}$

• Under certain conditions, a normal approximation is possible, with variance:{{cite journal | last1=Díaz-Francés | first1=Eloísa | last2=Rubio | first2=Francisco J. | title=On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables | journal=Statistical Papers | publisher=Springer Science and Business Media LLC | volume=54 | issue=2 | date=2012-01-24 | issn=0932-5026 | doi=10.1007/s00362-012-0429-2 | pages=309–323| s2cid=122038290 }}

:$\backslash sigma\_z^2=\backslash frac\{\backslash mu\_x^2\}\{\backslash mu\_y^2\}\; \backslash left(\backslash frac\{\backslash sigma\_x^2\}\{\backslash mu\_x^2\}\; +\; \backslash frac\{\backslash sigma\_y^2\}\{\backslash mu\_y^2\}\backslash right)$

The above expression becomes more complicated when the variables X and Y are correlated. If $\backslash mu\_x\; =\; \backslash mu\_y\; =\; 0$ but $\backslash sigma\_X\; \backslash neq\; \backslash sigma\_Y$ and $\backslash rho\; \backslash neq\; 0$ the more general Cauchy distribution is obtained

: $p\_Z(z)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash frac\{\backslash beta\}\{(z-\backslash alpha)^2\; +\; \backslash beta^2\},$

where ρ is the correlation coefficient between X and Y and

: $\backslash alpha\; =\; \backslash rho\; \backslash frac\{\backslash sigma\_x\}\{\backslash sigma\_y\},$

: $\backslash beta\; =\; \backslash frac\{\backslash sigma\_x\}\{\backslash sigma\_y\}\; \backslash sqrt\{1-\backslash rho^2\}.$

The complex distribution has also been expressed with Kummer's confluent hypergeometric function or the Hermite function.

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

:$T\; \backslash sim\; \backslash frac\{\backslash mu\_x\; +\; \backslash mathbb\{N\}(0,\; \backslash sigma\_x^2\; )\}\{\backslash mu\_y\; +\; \backslash mathbb\{N\}(0,\; \backslash sigma\_y^2\; )\}$

= \frac{\mu_x + X}{\mu_y + Y}

= \frac{\mu_x}{\mu_y}\frac{1+ \frac{X}{\mu_x}}{1+ \frac{Y}{\mu_y}} .

Take logs to get

:$\backslash log\_e(T)\; =\; \backslash log\_e\; \backslash left(\backslash frac\{\backslash mu\_x\}\{\backslash mu\_y\}\; \backslash right)$

+ \log_e \left( 1+ \frac{X}{\mu_x} \right)

- \log_e \left( 1+ \frac{Y}{\mu_y} \right)

.

Since $\backslash log\_e(1+\backslash delta)\; =\; \backslash delta\; -\; \backslash frac\{\backslash delta^2\}\{2\}\; +\; \backslash frac\{\backslash delta^3\}\{3\}\; +\; \backslash cdots$ then asymptotically

:$\backslash log\_e(T)\; \backslash approx\; \backslash log\_e\; \backslash left(\backslash frac\{\backslash mu\_x\}\{\backslash mu\_y\}\; \backslash right)+\; \backslash frac\{X\}\{\backslash mu\_x\}\; -$

\frac{Y}{\mu_y}

\sim \log_e \left(\frac{\mu_x}{\mu_y} \right) + \mathbb{N} \left( 0, \frac{\sigma_x^2}{\mu_x^2} + \frac{\sigma_y^2}{\mu_y^2} \right)

.

Alternatively, Geary (1930) suggested that

: $t\; \backslash approx\; \backslash frac\{\backslash mu\_y\; T\; -\; \backslash mu\_x\}\{\backslash sqrt\{\backslash sigma\_y^2\; T^2\; -\; 2\backslash rho\; \backslash sigma\_x\; \backslash sigma\_y\; T\; +\; \backslash sigma\_x^2\}\}$

has approximately a standard Gaussian distribution:

This transformation has been called the Geary–Hinkley transformation; the approximation is good if Y is unlikely to assume negative values, basically $\backslash mu\_y\; >\; 3\backslash sigma\_y$.

{{original synthesis|section|date=November 2019}}

This is developed by Dale (Springer 1979 problem 4.28) and Hinkley 1969. Geary showed how the correlated ratio $z$ could be transformed into a near-Gaussian form and developed an approximation for $t$ 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:

:$z=\backslash frac\; \{x+\backslash mu\_x\}\{y+\backslash mu\_y\}$

in which $x,\; y$ are zero-mean correlated normal variables with variances $\backslash sigma\_x^2,\; \backslash sigma\_y^2$ and $X,\; Y$ have means $\backslash mu\_x,\; \backslash mu\_y.$

Write $x\text{'}=x-\backslash rho\; y\backslash sigma\_x\; /\backslash sigma\_y$ such that $x\text{'},\; y$ become uncorrelated and $x\text{'}$ has standard deviation

:$\backslash sigma\_x\text{'}\; =\; \backslash sigma\_x\; \backslash sqrt\; \{1-\; \backslash rho^2\}.$

The ratio:

:$z=\backslash frac\{x\text{'}\; +\; \backslash rho\; y\backslash sigma\_x/\backslash sigma\_y+\backslash mu\_x\}\{y+\backslash mu\_y\}$

is invariant under this transformation and retains the same pdf.

The $y$ term in the numerator appears to be made separable by expanding:

:$\{x\text{'}\; +\; \backslash rho\; y\backslash sigma\_x/\backslash sigma\_y+\backslash mu\_x\}\; =x\text{'}+\backslash mu\_x\; -\backslash rho\; \backslash mu\_y\; \backslash frac\{\backslash sigma\_x\}\{\backslash sigma\_y\}\; +\; \backslash rho\; (y+\backslash mu\_y)\backslash frac\{\backslash sigma\_x\}\{\backslash sigma\_y\}$

to get

:$z=\backslash frac\; \{x\text{'}+\backslash mu\_x\text{'}\}\{y+\backslash mu\_y\}\; +\; \backslash rho\; \backslash frac\{\; \backslash sigma\_x\}\{\backslash sigma\_y\}$

in which $\backslash mu\text{'}\_x=\backslash mu\_x\; -\; \backslash rho\; \backslash mu\_y\; \backslash frac\; \{\; \backslash sigma\_x\; \}\{\backslash sigma\_y\}$ 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 $z$ for correlated variables is found by inputting the modified parameters $\backslash sigma\_x\text{'},\; \backslash mu\_x\text{'},\; \backslash sigma\_y,\; \backslash mu\_y$ and $\backslash rho\text{'}=0$ into the Hinkley equation above which returns the pdf for the correlated ratio with a constant offset $-\; \backslash rho\; \backslash frac\{\backslash sigma\_x\}\{\backslash sigma\_y\}$ on $z$.

{{multiple image

| width = 300

| image1 = WikiPic4.jpg

| alt1 = Gaussian ratio contours

| caption1 = Contours of the correlated bivariate Gaussian distribution (not to scale) giving ratio x/y

| image2 =Ratiodist2.jpg

| alt2 = pdf of probability distribution ratio z

| caption2 = pdf of the Gaussian ratio z and a simulation (points) for

$\backslash sigma\_x=\; \backslash sigma\_y=1,\; \backslash mu\_x=0,\; \backslash mu\_y=0.5,\; \backslash rho=0.975$

}}

The figures above show an example of a positively correlated ratio with $\backslash sigma\_x=\; \backslash sigma\_y=1,\; \backslash mu\_x=0,\; \backslash mu\_y=0.5,\; \backslash rho\; =\; 0.975$ in which the shaded wedges represent the increment of area selected by given ratio $x/y\; \backslash in\; [r,\; r\; +\; \backslash delta]$ 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 $z\; =\; x/y\; \backslash approx\; 1$ the wedge has almost bypassed the main distribution mass altogether and this explains the local minimum in the theoretical pdf $p\_Z(x/y)$. Conversely as $x/y$ 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.{{Cite book|last1=Baxley|first1=R T|last2=Waldenhorst|first2=B T|last3=Acosta-Marum|first3=G| date=2010| title= 2010 IEEE Global Telecommunications Conference GLOBECOM 2010|pages=1–5|chapter-url=https://www.researchgate.net/publication/224210655|doi=10.1109/GLOCOM.2010.5683407|chapter=Complex Gaussian Ratio Distribution with Applications for Error Rate Calculation in Fading Channels with Imperfect CSI|isbn=978-1-4244-5636-9|s2cid=14100052}} and has since been extended to the nonzero-mean and nonsymmetric case.{{cite journal |last1=Sourisseau |first1=M. |last2=Wu |first2=H.-T. |last3=Zhou |first3=Z. |title=Asymptotic analysis of synchrosqueezing transform—toward statistical inference with nonlinear-type time-frequency analysis |journal=Annals of Statistics |date=October 2022 |volume=50 |issue=5 |pages=2694–2712 |doi=10.1214/22-AOS2203 |url=https://projecteuclid.org/journals/annals-of-statistics/volume-50/issue-5/Asymptotic-analysis-of-synchrosqueezing-transformtoward-statistical-inference-with-nonlinear-type/10.1214/22-AOS2203.short?tab=ArticleLinkReference|arxiv=1904.09534 }} In the correlated zero-mean case, the joint distribution of x, y is

:$f\_\{x,y\}(x,y)\; =\; \backslash frac\{1\}\{\backslash pi^2\; |\backslash Sigma|\}\; \backslash exp\; \backslash left\; (\; -\; \backslash begin\{bmatrix\}x\; \backslash \backslash \; y\; \backslash end\{bmatrix\}^H\; \backslash Sigma\; ^\{-1\}\backslash begin\{bmatrix\}x\; \backslash \backslash \; y\; \backslash end\{bmatrix\}\; \backslash right\; )$

where

:$\backslash Sigma\; =\; \backslash begin\{bmatrix\}$

\sigma_x^2 & \rho \sigma_x \sigma_y \\

\rho^* \sigma_x \sigma_y & \sigma_y^2 \end{bmatrix}, \;\; x=x_r+ix_i, \;\; y=y_r+iy_i

$(\backslash cdot)^H$ is an Hermitian transpose and

:$\backslash rho\; =\; \backslash rho\_r\; +i\; \backslash rho\_i$

= \operatorname{E} \bigg(\frac{xy^*}{\sigma_x \sigma_y} \bigg )\;\; \in \;\left |\mathbb{C} \right| \le 1

The PDF of $Z\; =\; X/Y$ is found to be

:

\Biggr ( \frac{|z|^2}{\sigma_x^2} + \frac{1}{\sigma_y^2} -2\frac{\rho_r z_r - \rho_i z_i}{\sigma_x \sigma_y} \Biggr)^{-2} \\

& = \frac{1-|\rho|^2}{\pi \sigma_x^2 \sigma_y^2 }

\Biggr ( \;\; \Biggr | \frac{z}{\sigma_x} - \frac{\rho^* }{\sigma_y} \Biggr |^2 +\frac{1-|\rho|^2}{\sigma_y^2} \Biggr)^{-2}

\end{align}

In the usual event that $\backslash sigma\_x\; =\; \backslash sigma\_y$ we get

:$f\_\{z\}(z\_r,z\_i)\; =\; \backslash frac\{1-|\backslash rho|^2\}\{\backslash pi\; \backslash left\; (\; \backslash ;\backslash ;\; |\; z\; -\; \backslash rho^*\; |^2\; +\; 1-|\backslash rho|^2\; \backslash right)^\{2\}\; \}$

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

File:cmplxratio.jpg

The graph shows the pdf of the ratio of two complex normal variables with a correlation coefficient of $\backslash rho\; =\; 0.7\; \backslash exp\; (i\; \backslash pi\; /4)$. The pdf peak occurs at roughly the complex conjugate of a scaled down $\backslash rho$.

== Ratio of log-normal ==

The ratio of independent or correlated log-normals is log-normal. This follows, because if $X\_1$ and $X\_2$ are log-normally distributed, then $\backslash ln(X\_1)$ and $\backslash ln(X\_2)$ 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, however, that $X\_1$ and $X\_2$ 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.

This is important for many applications requiring the ratio of random variables that must be positive, where joint distribution of $X\_1$ and $X\_2$ 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 $X\_i$ is the result of an accumulation of many small percentage changes and must be positive and approximately log-normally distributed.Of course, any invocation of a central limit theorem assumes suitable, commonly met regularity conditions, e.g., finite variance.

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

: $p\_X(x)\; =\; \backslash begin\{cases\}$

1 & 0 < x < 1 \\

0 & \text{otherwise}

\end{cases}

the ratio distribution becomes

: $p\_Z(z)\; =\; \backslash begin\{cases\}$

1/2 \qquad & 0 < z < 1 \\

\frac{1}{2z^2} \qquad & z \geq 1 \\

0 \qquad & \text{otherwise}

\end{cases}

If two independent random variables, X and Y each follow a Cauchy distribution with median equal to zero and shape factor $a$

: $p\_X(x|a)\; =\; \backslash frac\{a\}\{\backslash pi\; (a^2\; +\; x^2)\}$

then the ratio distribution for the random variable $Z\; =\; X/Y$ is{{Cite journal

| title = An Introduction to the Algebra of Random Variables

| last = Kermond

| first = John

| journal = Mathematical Association of Victoria 47th Annual Conference Proceedings – New Curriculum. New Opportunities

| publisher = The Mathematical Association of Victoria

| year = 2010

| isbn = 978-1-876949-50-1

| pages = 1–16

}}

: $p\_Z(z|a)\; =\; \backslash frac\{1\}\{\backslash pi^2(z^2-1)\}\; \backslash ln(z^2).$

This distribution does not depend on $a$ 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 $W=XY$:

: $p\_W(w|a)\; =\; \backslash frac\{a^2\}\{\backslash pi^2(w^2-a^4)\}\; \backslash ln\; \backslash left(\backslash frac\{w^2\}\{a^4\}\backslash right).$

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

1. The ratio distribution for the random variable $Z\; =\; X/Y$ is $$p\_Z(z|a,b)\; =\; \backslash frac\{ab\}\{\backslash pi^2(b^2z^2-a^2)\}\; \backslash ln\; \backslash left(\backslash frac\{b^2\; z^2\}\{a^2\}\backslash right).$$
2. The product distribution for the random variable $W\; =\; XY$ is $$p\_W(w|a,b)\; =\; \backslash frac\{ab\}\{\backslash pi^2(w^2-a^2b^2)\}\; \backslash ln\; \backslash left(\backslash frac\{w^2\}\{a^2b^2\}\backslash right).$$

The result for the ratio distribution can be obtained from the product distribution by replacing $b$ with $\backslash frac\{1\}\{b\}.$

{{main|Slash distribution}}

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

:$p\_Z(z)\; =\; \backslash begin\{cases\}$

\left[ \varphi(0) - \varphi(z) \right] / z^2 \quad & z \ne 0 \\

\varphi(0) / 2 \quad & z = 0, \\

\end{cases}

where φ(z) is the probability density function of the standard normal distribution.{{cite web|url=http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/slappf.htm|title=SLAPPF|publisher=Statistical Engineering Division, National Institute of Science and Technology|access-date=2009-07-02}}

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 $f\_\backslash chi\; (x,k)$

= \frac {x^ {\frac{k}{2}-1} e^{-x/2} } { 2^{k/2} \Gamma(k/2) }. Then

: $\backslash frac\{\; G\; \}\{\; \backslash sqrt\{\; Y\; /\; m\; \}\; \}\; \backslash sim\; t\_m$ the Student's t distribution

: $\backslash frac\{\; Y\; /\; m\; \}\{\; Z\; /\; n\; \}\; =\; F\_\{\; m,\; n\; \}$ i.e. Fisher's F-test distribution

: $\backslash frac\{\; Y\; \}\{\; Y\; +\; Z\; \}\; \backslash sim\; \backslash beta(\; \backslash tfrac\{m\}\{2\},\backslash tfrac\{n\}\{2\}\; )$ the beta distribution

: $\backslash ;\backslash ;\backslash frac\{\; Y\; \}\{\; Z\; \}\; \backslash sim\; \backslash beta\text{'}(\; \backslash tfrac\{m\}\{2\},\backslash tfrac\{n\}\{2\}\; )$ the standard beta prime distribution

If $V\_1\; \backslash sim\; \{\backslash chi\text{'}\}\_\{k\_1\}^2(\backslash lambda)$, a noncentral chi-squared distribution, and $V\_2\; \backslash sim\; \{\backslash chi\text{'}\}\_\{k\_2\}^2(0)$ and $V\_1$ is independent of $V\_2$ then

: $\backslash frac\{V\_1/k\_1\}\{V\_2/k\_2\}\; \backslash sim\; F\text{'}\_\{k\_1,k\_2\}(\backslash lambda)$, a noncentral F-distribution.

$\backslash frac\{m\}\{n\}\; F\text{'}\_\{m,n\}\; =\; \backslash beta\text{'}(\; \backslash tfrac\{m\}\{2\},\backslash tfrac\{n\}\{2\})\; \backslash text\{\; or\; \}$

F'_{m,n} = \beta'( \tfrac{m}{2},\tfrac{n}{2} ,1,\tfrac{n }{m})

defines $F\text{'}\_\{m,n\}$, 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

: $F\_\{3,4\}(6.59)\; =\; \backslash int\_\{6.59\}^\backslash infty\; \backslash beta\text{'}(x;\; \backslash tfrac\{m\}\{2\},\backslash tfrac\{n\}\{2\},1,\backslash tfrac\{n\}\{m\}\; )\; dx\; =\; 0.05$

For gamma distributions U and V with arbitrary shape parameters α₁ and α₂ and their scale parameters both set to unity, that is, $U\; \backslash sim\; \backslash Gamma\; (\; \backslash alpha\_1\; ,\; 1),\; V\; \backslash sim\; \backslash Gamma(\backslash alpha\_2,\; 1)$, where $$

\Gamma (x;\alpha,1) = \frac { x^{\alpha-1} e^{-x}}{\Gamma(\alpha)} , then

: $\backslash frac\{\; U\; \}\{\; U\; +\; V\; \}\; \backslash sim\; \backslash beta(\; \backslash alpha\_1,\; \backslash alpha\_2\; ),\; \backslash qquad\; \backslash text\{\; expectation\; \}\; =\; \backslash frac\{\backslash alpha\_1\}\{\backslash alpha\_1\; +\; \backslash alpha\_2\; \}$

: $\backslash frac\{U\}\{V\}\; \backslash sim\; \backslash beta\text{'}(\backslash alpha\_1,\backslash alpha\_2),\; \backslash qquad\; \backslash qquad\; \backslash text\{\; expectation\; \}\; =\; \backslash frac\{\backslash alpha\_1\}\{\; \backslash alpha\_2\; -1\},\; \backslash ;\; \backslash alpha\_2\; >\; 1$

: $\backslash frac\{V\}\{U\}\; \backslash sim\; \backslash beta\text{'}(\backslash alpha\_2,\; \backslash alpha\_1),\; \backslash qquad\; \backslash qquad\; \backslash text\{\; expectation\; \}\; =\; \backslash frac\{\backslash alpha\_2\}\{\; \backslash alpha\_1\; -1\},\; \backslash ;\; \backslash alpha\_1\; >\; 1$

If $U\; \backslash sim\; \backslash Gamma\; (x;\backslash alpha,1)$, then $\backslash theta\; U\; \backslash sim\; \backslash Gamma\; (x;\backslash alpha,\backslash theta)\; =\; \backslash frac\; \{\; x^\{\backslash alpha-1\}\; e^\{-\; \backslash frac\{x\}\{\backslash theta\}\}\}\{\; \backslash theta^k\; \backslash Gamma(\backslash alpha)\}$. Note that here θ is a scale parameter, rather than a rate parameter.

If $U\; \backslash sim\; \backslash Gamma(\backslash alpha\_1,\; \backslash theta\_1\; ),\backslash ;\; V\; \backslash sim\; \backslash Gamma(\backslash alpha\_2,\; \backslash theta\_2\; )$, then by rescaling the $\backslash theta$ parameter to unity we have

: $\backslash frac\; \{\backslash frac\; \{U\}\{\backslash theta\_1\}\}\; \{\; \backslash frac\; \{U\}\{\backslash theta\_1\}\; +\; \backslash frac\; \{V\}\{\backslash theta\_2\}\}$

= \frac{ \theta_2 U }{ \theta_2 U + \theta_1 V } \sim \beta( \alpha_1, \alpha_2 )

: $\backslash frac\; \{\backslash frac\; \{U\}\{\backslash theta\_1\}\}\; \{\; \backslash frac\; \{V\}\{\backslash theta\_2\}\}$

= \frac{ \theta_2 }{ \theta_1 } \frac{U }{ V }\sim \beta'( \alpha_1, \alpha_2 )

Thus

: $\backslash frac\; \{U\}\{V\}\; \backslash sim\; \backslash beta\text{'}(\; \backslash alpha\_1,\; \backslash alpha\_2,\; 1,\; \backslash frac\{\backslash theta\_1\; \}\{\; \backslash theta\_2\; \}\; )\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash operatorname\{E\}\; \backslash left[\; \backslash frac\; \{U\}\{V\}\; \backslash right]\; =\; \backslash frac\{\backslash theta\_1\; \}\{\; \backslash theta\_2\; \}\; \backslash frac\{\backslash alpha\_1\}\{\backslash alpha\_2\; -\; 1\; \}$

in which $\backslash beta\text{'}(\backslash alpha,\backslash beta,p,q)$ represents the generalised beta prime distribution.

In the foregoing it is apparent that if $X\; \backslash sim\; \backslash beta\text{'}(\; \backslash alpha\_1,\; \backslash alpha\_2,\; 1,\; 1\; )\; \backslash equiv\; \backslash beta\text{'}(\; \backslash alpha\_1,\; \backslash alpha\_2\; )$ then $\backslash theta\; X\; \backslash sim\; \backslash beta\text{'}(\; \backslash alpha\_1,\; \backslash alpha\_2,\; 1,\; \backslash theta\; )$. More explicitly, since

: $\backslash beta\text{'}(x;\; \backslash alpha\_1,\; \backslash alpha\_2,\; 1,\; R\; )\; =\; \backslash frac\{1\}\{R\}\; \backslash beta\text{'}\; (\backslash frac\{x\}\{R\}\; ;\; \backslash alpha\_1,\; \backslash alpha\_2)$

if $U\; \backslash sim\; \backslash Gamma(\backslash alpha\_1,\; \backslash theta\_1\; ),\; V\; \backslash sim\; \backslash Gamma(\backslash alpha\_2,\; \backslash theta\_2\; )$

then

:$\backslash frac\; \{U\}\{V\}\; \backslash sim\; \backslash frac\{1\}\{R\}\; \backslash beta\text{'}\; (\; \backslash frac\{x\}\{R\}\; ;\; \backslash alpha\_1,\; \backslash alpha\_2\; )$

= \frac { \left(\frac{x}{R} \right)^{\alpha_1-1} } {\left(1+\frac{x}{R} \right)^{\alpha_1+\alpha_2}} \cdot

\frac {1} { \;R\;B( \alpha_1, \alpha_2 )}, \;\; x \ge 0

where

: $R\; =\; \backslash frac\; \{\backslash theta\_1\}\{\backslash theta\_2\},\; \backslash ;\; \backslash ;\backslash ;$

B( \alpha_1, \alpha_2 ) = \frac {\Gamma(\alpha_1) \Gamma(\alpha_2)} {\Gamma(\alpha_1 + \alpha_2)}

If X, Y are independent samples from the Rayleigh distribution $f\_r(r)\; =\; (r/\backslash sigma^2)\; e^\; \{-r^2/2\backslash sigma^2\},\; \backslash ;\backslash ;\; r\; \backslash ge\; 0$, the ratio Z = X/Y follows the distribution{{Cite journal|last=Hamedani|first=G. G.|date=Oct 2013| title=Characterizations of Distribution of Ratio of Rayleigh Random Variables|journal=Pakistan Journal of Statistics|volume=29| issue=4|pages=369–376}}

:$f\_z(z)\; =\; \backslash frac\{2\; z\}\{\; (1\; +\; z^2\; )^2\; \},\; \backslash ;\backslash ;\; z\; \backslash ge\; 0$

and has cdf

:$F\_z(z)\; =\; 1\; -\; \backslash frac\{1\}\{\; 1\; +\; z^2\; \}\; =\; \backslash frac\{z^2\}\{\; 1\; +\; z^2\; \},\; \backslash ;\backslash ;\backslash ;\; z\; \backslash ge\; 0$

The Rayleigh distribution has scaling as its only parameter. The distribution of $Z\; =\; \backslash alpha\; X/Y$ follows

:$f\_z(z,\backslash alpha)\; =\; \backslash frac\{2\; \backslash alpha\; z\}\{\; (\backslash alpha\; +\; z^2\; )^2\; \},\; \backslash ;\backslash ;\; z\; >\; 0$

and has cdf

:$F\_z(z,\; \backslash alpha)\; =\; \backslash frac\{\; z^2\; \}\{\; \backslash alpha\; +\; z^2\; \},\; \backslash ;\backslash ;\backslash ;\; z\; \backslash ge\; 0$

The generalized gamma distribution is

: $f(x;a,d,r)=\backslash frac\{r\}\{\backslash Gamma(d/r)\; a^d\; \}\; x^\{d-1\}\; e^\{-(x/a)^r\}\; \backslash ;\; x\; \backslash ge\; 0;\; \backslash ;\backslash ;\; a,\; \backslash ;\; d,\; \backslash ;r\; >\; 0$

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.

: If $U\; \backslash sim\; f(x;a\_1,d\_1,r),\; \backslash ;\; \backslash ;\; V\; \backslash sim\; f(x;a\_2,d\_2,r)\; \backslash text\{\; are\; independent,\; and\; \}\; W\; =\; U/V$

: then{{Cite journal

| title = A note on the generalized (positive) Cauchy distribution

| last1 = Raja Rao | first1 = B.

| last2 = Garg. | first2 = M. L.

| journal = Canadian Mathematical Bulletin

| volume = 12

| issue = 6

| year = 1969

| doi = 10.4153/CMB-1969-114-2 | doi-access=free

| pages = 865–868

}} $g(w)\; =\; \backslash frac\{r\; \backslash left\; (\; \backslash frac\; \{a\_1\}\{a\_2\}\; \backslash right\; )^\{d\_2\}\; \}\{B\; \backslash left\; (\; \backslash frac\{d\_1\}\{r\},\; \backslash frac\{d\_2\}\{r\}\; \backslash right\; )\; \}\; \backslash frac\{w^\{-d\_2\; -1\}\}\{\; \backslash left(\; 1\; +\; \backslash left(\; \backslash frac\{a\_2\}\{a\_1\}\; \backslash right)^\{-r\}\; w^\{-r\}\; \backslash right)\; ^\; \backslash frac\{d\_1+d\_2\}\{r\}\; \}\; ,\; \backslash ;\; \backslash ;\; w>0$

:where $B(u,v)\; =\; \backslash frac\{\backslash Gamma(u)\; \backslash Gamma(v)\}\{\backslash Gamma(u+v)\}$

In the ratios above, Gamma samples, U, V may have differing sample sizes $\backslash alpha\_1,\; \backslash alpha\_2$ but must be drawn from the same distribution $\backslash frac\; \{\; x^\{\backslash alpha-1\}\; e^\{-\; \backslash frac\{x\}\{\backslash theta\}\}\}\{\; \backslash theta^k\; \backslash Gamma(\backslash alpha)\}$ with equal scaling $\backslash theta$.

In situations where U and V are differently scaled, a variables transformation allows the modified random ratio pdf to be determined. Let $X\; =\; \backslash frac\; \{U\}\; \{\; U\; +\; V\}\; =\; \backslash frac\; \{1\}\; \{\; 1\; +\; B\}$ where $U\; \backslash sim\; \backslash Gamma(\backslash alpha\_1,\backslash theta),\; V\; \backslash sim\; \backslash Gamma(\backslash alpha\_2,\backslash theta),\; \backslash theta$ arbitrary and, from above, $X\; \backslash sim\; Beta(\backslash alpha\_1,\; \backslash alpha\_2),\; B\; =\; V/U\; \backslash sim\; Beta\text{'}(\backslash alpha\_2,\; \backslash alpha\_1)$.

Rescale V arbitrarily, defining $Y\; \backslash sim\; \backslash frac\; \{U\}\; \{\; U\; +\; \backslash varphi\; V\}\; =\; \backslash frac\; \{1\}\; \{\; 1\; +\; \backslash varphi\; B\},\; \backslash ;\backslash ;\; 0\; \backslash le\; \backslash varphi\; \backslash le\; \backslash infty$

We have $B\; =\; \backslash frac\{1-X\}\{X\}$ and substitution into Y gives $Y\; =\; \backslash frac\; \{X\}\{\backslash varphi\; +\; (1-\backslash varphi)X\},\; dY/dX\; =\; \backslash frac\; \{\backslash varphi\}\{(\backslash varphi\; +\; (1-\backslash varphi)X)^2\}$

Transforming X to Y gives $f\_Y(Y)\; =\; \backslash frac\{f\_X\; (X)\; \}$

Noting $X\; =\; \backslash frac\; \{\backslash varphi\; Y\}\{\; 1-(1\; -\; \backslash varphi)Y\}$ we finally have

: $f\_Y(Y,\; \backslash varphi)\; =\; \backslash frac\{\backslash varphi\; \}\; \{\; [1\; -\; (1\; -\; \backslash varphi)Y]^2\}\; \backslash beta\; \backslash left\; (\backslash frac\; \{\backslash varphi\; Y\}\{\; 1\; -\; (1-\backslash varphi)\; Y\},\; \backslash alpha\_1,\; \backslash alpha\_2\; \backslash right),\; \backslash ;\backslash ;\backslash ;\; 0\; \backslash le\; Y\; \backslash le\; 1$

Thus, if $U\; \backslash sim\; \backslash Gamma(\backslash alpha\_1,\backslash theta\_1)$ and $V\; \backslash sim\; \backslash Gamma(\backslash alpha\_2,\backslash theta\_2)$

then $Y\; =\; \backslash frac\; \{U\}\; \{\; U\; +\; V\}$ is distributed as $f\_Y(Y,\; \backslash varphi)$ with $\backslash varphi\; =\; \backslash frac\; \{\backslash theta\_2\}\{\backslash theta\_1\}$

The distribution of Y is limited here to the interval [0,1]. It can be generalized by scaling such that if $Y\; \backslash sim\; f\_Y(Y,\backslash varphi)$ then

: $\backslash Theta\; Y\; \backslash sim\; f\_Y(\; Y,\backslash varphi,\; \backslash Theta)$

where $f\_Y(\; Y,\backslash varphi,\; \backslash Theta)\; =\; \backslash frac\{\backslash varphi\; /\; \backslash Theta\; \}\; \{\; [1\; -\; (1\; -\; \backslash varphi)Y\; /\; \backslash Theta]^2\}\; \backslash beta\; \backslash left\; (\backslash frac\; \{\backslash varphi\; Y\; /\; \backslash Theta\}\{\; 1\; -\; (1-\backslash varphi)\; Y\; /\; \backslash Theta\},\; \backslash alpha\_1,\; \backslash alpha\_2\; \backslash right),\; \backslash ;\backslash ;\backslash ;\; 0\; \backslash le\; Y\; \backslash le\; \backslash Theta$

: $\backslash Theta\; Y$ is then a sample from $\backslash frac\; \{\backslash Theta\; U\}\; \{\; U\; +\; \backslash varphi\; V\}$

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

: If $X\; \backslash sim\; \backslash beta\; (\backslash alpha,\backslash beta)$ then $\backslash mathbf\; x\; =\; \backslash frac\{X\}\{1-X\}\; \backslash sim\; \backslash beta\text{'}(\backslash alpha,\backslash beta)$

: If $\backslash mathbf\; Y\; \backslash sim\; \backslash beta\text{'}\; (\backslash alpha,\backslash beta)$ then $y\; =\; \backslash frac\{1\}\{\backslash mathbf\; Y\}\; \backslash sim\; \backslash beta\text{'}(\backslash beta,\backslash alpha)$

combining the latter two equations yields

: If $X\; \backslash sim\; \backslash beta\; (\backslash alpha,\backslash beta)$ then $\backslash mathbf\; x\; =\; \backslash frac\{1\}\{X\}\; -1\; \backslash sim\; \backslash beta\text{'}(\backslash beta,\backslash alpha)$.

:

: If $\backslash mathbf\; Y\; \backslash sim\; \backslash beta\text{'}\; (\backslash alpha,\backslash beta)$ then $y\; =\; \backslash frac\{\backslash mathbf\; Y\}\{1\; +\; \backslash mathbf\; Y\}\; \backslash sim\; \backslash beta(\backslash alpha,\backslash beta)$

Corollary

: $\backslash frac\{1\}\{1\; +\; \backslash mathbf\; Y\}$

= \frac{ \mathbf Y ^ {-1}}{\mathbf Y^{-1} + 1} \sim \beta(\beta,\alpha)

: $1+\backslash mathbf\; Y\; \backslash sim\; \backslash \{\; \backslash ;\; \backslash beta(\backslash beta,\backslash alpha)\; \backslash ;\; \backslash \}\; ^\{-1\}$, the distribution of the reciprocals of $\backslash beta(\backslash beta,\backslash alpha)$ samples.

If $U\; \backslash sim\; \backslash Gamma\; (\; \backslash alpha\; ,\; 1),\; V\; \backslash sim\; \backslash Gamma(\backslash beta,\; 1)$ then $\backslash frac\{U\}\{V\}\; \backslash sim\; \backslash beta\text{'}\; (\; \backslash alpha,\; \backslash beta\; )$ and

: $\backslash frac\{U\; /\; V\}\{1+U\; /\; V\}\; =\; \backslash frac\{U\}\{V\; +\; U\; \}\; \backslash sim\; \backslash beta(\backslash alpha,\backslash beta)$

Further results can be found in the Inverse distribution article.

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

This result was derived by Katz et al.Katz D. et al.(1978) Obtaining confidence intervals for the risk ratio in cohort studies. Biometrics 34:469–474

Suppose $X\; \backslash sim\; \backslash text\{Binomial\}(n,p\_1)$ and

$Y\; \backslash sim\; \backslash text\{Binomial\}(m,p\_2)$ and $X$, $Y$ are independent. Let $T=\backslash frac\{X/n\}\{Y/m\}$.

Then $\backslash log(T)$ is approximately normally distributed with mean $\backslash log(p\_1/p\_2)$ and variance $\backslash frac\{(1/p\_1)-1\}\{n\}+\backslash frac\{(1/p\_2)-1\}\{m\}$.

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|date=November 2019}}

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 $e^\{-\backslash lambda\}$, the generic pdf of a left truncated Poisson distribution is

:$\backslash tilde\; p\_x(x;\backslash lambda)=\; \backslash frac\; \{1\}\{1-e^\{-\backslash lambda\}\; \}$

{ \frac{e^{-\lambda} \lambda^{x}}{x!} }, \;\;\; x \in 1,2,3, \cdots

which sums to unity. Following Cohen,{{Cite journal|last=Cohen|first=A Clifford|date=June 1960|title=Estimating the Parameter in a Conditional Poisson Distribution|journal=Biometrics| volume=60|issue=2|pages=203–211|doi=10.2307/2527552 | jstor=2527552}} for n independent trials, the multidimensional truncated pdf is

:$\backslash tilde\; p(x\_1,\; x\_2,\; \backslash dots\; ,x\_n;\backslash lambda)=\; \backslash frac\{1\}\{(1-e^\{-\backslash lambda\})^\{n\}\; \}$

\prod_{i=1}^n{ \frac{e^{-\lambda} \lambda^{x_i}}{x_i!} }, \;\;\; x_i \in 1,2,3, \cdots

and the log likelihood becomes

:$L\; =\; \backslash ln\; (\backslash tilde\; p)\; =-n\backslash ln\; (1-e^\{-\backslash lambda\})$

-n \lambda + \ln(\lambda) \sum_1^n x_i - \ln \prod_1^n (x_i!), \;\;\; x_i \in 1,2,3, \cdots

On differentiation we get

:$dL/d\backslash lambda\; =\; \backslash frac\{-n\}\{\; 1-e^\{-\backslash lambda\}\}\; +\; \backslash frac\{1\}\{\backslash lambda\}\backslash sum\_\{i=1\}^n\; x\_i$

and setting to zero gives the maximum likelihood estimate $\backslash hat\; \backslash lambda\_\{ML\}$

:$\backslash frac\{\backslash hat\; \backslash lambda\_\{ML\}\}\{\; 1-e^\{-\backslash hat\; \backslash lambda\_\{ML\}\; \}\}\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; x\_i\; =\; \backslash bar\; x$

Note that as $\backslash hat\; \backslash lambda\; \backslash to\; 0$ then $\backslash bar\; x\; \backslash to\; 1$ so the truncated maximum likelihood $\backslash lambda$ estimate, though correct for both truncated and untruncated distributions, gives a truncated mean $\backslash bar\; x$ value which is highly biassed relative to the untruncated one. Nevertheless it appears that $\backslash bar\; x$ is a sufficient statistic for $\backslash lambda$ since $\backslash hat\; \backslash lambda\_\{ML\}$ depends on the data only through the sample mean $\backslash bar\; x\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; x\_i$ 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 $\backslash lambda$ is valid over the whole range $0\; \backslash le\; \backslash lambda\; \backslash le\; \backslash infty;\; \backslash ;\; 1\; \backslash le\; \backslash bar\; x\; \backslash le\; \backslash infty$.

:

which compares with the non-truncated version which is simply $\backslash hat\; \backslash lambda\; =\; \backslash bar\; x$. Taking the ratio $R\; =\; \backslash hat\; \backslash lambda\_X\; /\; \backslash hat\; \backslash lambda\_Y$ is a valid operation even though $\backslash hat\; \backslash lambda\_X$ may use a non-truncated model while $\backslash hat\; \backslash lambda\_Y$ has a left-truncated one.

The asymptotic large-$n\backslash lambda\; \backslash text\{\; variance\; of\; \}\backslash hat\; \backslash lambda$ (and Cramér–Rao bound) is

:$\backslash mathbb\{Var\}\; (\; \backslash hat\; \backslash lambda\; )\; \backslash ge\; -\; \backslash left(\; \backslash mathbb\{E\}\backslash left[\; \backslash frac\{\backslash delta\; ^2\; L\; \}\{\; \backslash delta\; \backslash lambda^2\; \}\; \backslash right]\_\{\backslash lambda=\backslash hat\; \backslash lambda\}\; \backslash right)\; ^\{-1\}$

in which substituting L gives

:$\backslash frac\{\backslash delta\; ^2\; L\; \}\{\; \backslash delta\; \backslash lambda^2\; \}\; =$

-n \left[ \frac{ \bar x}{\lambda ^2 } - \frac{e^{-\lambda}}{(1-e^{-\lambda})^2 } \right]

Then substituting $\backslash bar\; x$ from the equation above, we get Cohen's variance estimate

:$\backslash mathbb\{Var\}\; (\; \backslash hat\; \backslash lambda\; )\; \backslash ge\; \backslash frac\{\; \backslash hat\backslash lambda\}\{n\}\; \backslash frac\; \{\; (1-e^\{-\backslash hat\backslash lambda\})^2\; \}\{\; 1\; -\; (\backslash hat\backslash lambda\; +\; 1)\; e^\{-\backslash hat\backslash lambda\}\; \}$

The variance of the point estimate of the mean $\backslash lambda$, on the basis of n trials, decreases asymptotically to zero as n increases to infinity. For small $\backslash lambda$ it diverges from the truncated pdf variance in Springael{{Cite web|url=https://repository.uantwerpen.be/docman/irua/b66c5a/4743e891.pdf|title=On the sum of independent zero-truncated Poisson random variables|last=Springael|first=Johan|date=2006|website=University of Antwerp, Faculty of Business and Economics}} for example, who quotes a variance of

:$\backslash mathbb\; \{Var\}\; (\; \backslash lambda)\; =\; \backslash frac\; \{\backslash lambda\; /\; n\}\{1-e^\{-\backslash lambda\}\}\; \backslash left\; [\; 1\; -\; \backslash frac\{\backslash lambda\; e^\{-\backslash lambda\}\; \}\{1-e^\{-\backslash lambda\}\}\backslash right]$

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, $\backslash mathbb\; \{Var\}\; (\; \backslash hat\; \backslash lambda)\; /\; \backslash mathbb\; \{Var\}\; (\; \backslash lambda)$, ranges from 1 for large $\backslash lambda$ (100% efficient) up to 2 as $\backslash lambda$ 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{{Cite journal|last1=Dietz|first1=Ekkehart|last2=Bohning|first2=Dankmar| date=2000 |title=On Estimation of the Poisson Parameter in Zero-Modified Poisson Models|journal=Computational Statistics & Data Analysis |volume=34|issue=4|pages=441–459|doi=10.1016/S0167-9473(99)00111-5}} and there is a Zero-truncated Poisson distribution Wikipedia entry.

This distribution is the ratio of two Laplace distributions.Bindu P and Sangita K (2015) Double Lomax distribution and its applications. Statistica LXXV (3) 331–342 Let X and Y be standard Laplace identically distributed random variables and let z = X / Y. Then the probability distribution of z is

: $f(\; x\; )\; =\; \backslash frac\{\; 1\; \}\{\; 2\; (\; 1\; +\; |z|\; )^2\; \}$

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.

: $E(\; Z^a\; )\; =\; \backslash frac\{\; \backslash Gamma(\; 1\; +\; a\; )\; \}\{\; \backslash Gamma(\; 1\; -\; a\; )\; \}$

where Γ is the gamma function.

Ratio distributions also appear in multivariate analysis.{{Cite journal|last1=Brennan|first1=L E| last2=Reed| first2=I S|date=January 1982|title=An Adaptive Array Signal Processing Algorithm for Communications|journal=IEEE Transactions on Aerospace and Electronic Systems|volume=AES-18 No 1|pages=124–130|bibcode=1982ITAES..18..124B| doi=10.1109/TAES.1982.309212 |s2cid=45721922}} If the random matrices X and Y follow a Wishart distribution then the ratio of the determinants

: $\backslash varphi\; =\; |\backslash mathbf\{X\}|/|\backslash mathbf\{Y\}|$

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

: $\backslash Lambda\; =$

has a Wilks' lambda distribution.

In relation to Wishart matrix distributions if $S\; \backslash sim\; W\_p(\backslash Sigma,\; \backslash nu\; +\; 1)$ is a sample Wishart matrix and vector $V$ is arbitrary, but statistically independent, corollary 3.2.9 of Muirhead states

:$\backslash frac\{\; V^T\; S\; V\}\{\; V^T\; \backslash Sigma\; V\; \}\; \backslash sim\; \backslash chi^2\_\{\backslash nu\; \}$

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

:$\backslash frac\{\; V^T\; \backslash Sigma^\{-1\}\; V\}\{\; V^T\; S^\{-1\}\; V\; \}\; \backslash sim\; \backslash chi^2\_\{\backslash nu-p+1\}$

which is Theorem 3.2.12 of Muirhead {{Cite book|last=Muirhead|first=Robb|title=Aspects of Multivariate Statistical Theory|publisher=Wiley|year=1982|location=USA|pages=96; Theorem 3.2.12}}

• Relationships among probability distributions
• Inverse distribution (also known as reciprocal distribution)
• Product distribution
• Ratio estimator
• Slash distribution

{{reflist|group=note}}

{{Reflist}}

• [http://mathworld.wolfram.com/RatioDistribution.html Ratio Distribution] at MathWorld
• [http://mathworld.wolfram.com/NormalRatioDistribution.html Normal Ratio Distribution] at MathWorld
• [http://www.mathpages.com/home/kmath042/kmath042.htm Ratio Distributions] at MathPages

Category:Algebra of random variables

Category:Statistical ratios

Category:Types of probability distributions