81.151.118.231 at 01:31, 30 January 2014

2014-01-30T01:31:40Z

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{{citations|date=April 2013}}

{{Distinguish|Inverse distribution function}}
In [[probability theory]] and [[statistics]], an '''inverse distribution''' is the distribution of the [[multiplicative inverse|reciprocal]] of a random variable. Inverse distributions arise in particular in the [[Bayesian inference|Bayesian]] context of [[prior distribution]]s and [[posterior distribution]]s for [[scale parameter]]s. In the [[algebra of random variables]], inverse distributions are special cases of the class of [[ratio distribution]]s, in which the numerator random variable has a [[degenerate distribution]].

==Relation to original distribution==

In general, given the [[probability distribution]] of a random variable ''X'' with strictly positive support, it is possible to find the distribution of the reciprocal, ''Y'' = 1 / ''X''. If the distribution of ''X'' is [[Continuous probability distribution|continuous]] with [[Probability density function|density function]] ''f''(''x'') and [[cumulative distribution function]] ''F''(''x''), then the cumulative distribution function, ''G''(''y''), of the reciprocal is found by noting that

:<math> G(y) = \Pr(Y \leq y) = \Pr\left(X \geq \frac{1}{y}\right) = 1-\Pr\left(X<\frac{1}{y}\right) = 1 - F\left( \frac{ 1 }{ y } \right).</math>

Then the density function of ''Y'' is found as the derivative of the cumulative distribution function:

: <math> g(y) = \frac{ 1 }{ y^2 } f\left( \frac{ 1 }{ y } \right) . </math>

==Examples==
===Reciprocal distribution===
The [[reciprocal distribution]] has a density function of the form.<ref name=Hamming1970>[[Richard Hamming|Hamming R. W.]] (1970) [http://lucent.com/bstj/vol49-1970/articles/bstj49-8-1609.pdf "On the distribution of numbers"], ''The Bell System Technical Journal'' 49(8) 1609–1625</ref>

:<math>f(x) \propto x^{-1} \quad \text{ for } 0<a<x<b, </math>

where <math>\propto \!\,</math> means [[Proportionality (mathematics)|"is proportional to"]].
It follows that the inverse distribution in this case is of the form
:<math>g(y) \propto y^{-1} \quad \text{ for } 0\le b^{-1}<y< a^{-1}, </math>
which is again a reciprocal distribution.

===Inverse uniform distribution===
{{Probability distribution|
name =Inverse uniform distribution|
type =density|
pdf_image =|
cdf_image =|
parameters =<math> 0 < a < b, \quad a, b \in \R</math>|
support =<math> [ b^{-1} , a^{-1} ] </math>|
pdf =<math> y^{-2} \frac{ 1 }{ b-a } </math>|
cdf =<math> \frac{ b - y^{-1} }{ b - a } </math>|
mean = |
median = <math> \frac{ 2}{ a+b }</math>
| variance =
| skewness =
| kurtosis =
| entropy =
| mgf =
| char =
| pgf =
| fisher =
}}

If the original random variable ''X'' is [[uniform distribution (continuous)|uniformly distributed]] on the interval (''a'',''b''), where ''a''>0, then the reciprocal variable ''Y'' = 1 / ''X'' has the reciprocal distribution which takes values in the range (''b''-1 ,''a''-1), and the probability density function in this range is

: <math> g( y ) = y^{-2} \frac{ 1 }{ b-a } ,</math>

and is zero elsewhere.

The cumulative distribution function of the reciprocal, within the same range, is

: <math> G( y ) = \frac{ b - y^{-1} }{ b - a } .</math>

===Inverse t distribution===

Let ''X'' be a [[Student's t-distribution|''t'' distributed]] random variate with ''k'' [[degrees of freedom]]. Then its density function is

: <math> f( x ) = \frac{ 1 }{ \sqrt{ k \pi } } \frac{ \Gamma\left( \frac{ k + 1 }{ 2 } \right) }{ \Gamma\left( \frac{ k }{ 2 } \right) } \frac{ 1 }{ \left( 1 + \frac{ x^2 }{ k } \right)^{ \frac{ 1 + k }{ 2 } } } .</math>

The density of ''Y'' = 1 / ''X'' is

: <math> g( y ) = \frac{ 1 }{ \sqrt{ k \pi } } \frac{ \Gamma\left( \frac{ k + 1 }{ 2 } \right) }{ \Gamma\left( \frac{ k }{ 2 } \right) } \frac{ 1 }{ y^2 \left( 1 + \frac{ 1 }{ y^2 k } \right)^{ \frac{ 1 + k }{ 2 } } } .</math>

With ''k'' = 1, the distributions of ''X'' and 1 / ''X'' are identical. If ''k'' > 1 then the distribution of 1 / ''X'' is [[bimodal]].{{cn|date=April 2013}}

===Reciprocal normal distribution===

If ''X'' is a standard [[normally distributed]] variable then the distribution of 1/''X'' is bimodal,<ref name=Johnson>{{cite book
| last1 = Johnson | first1 = Norman L.
| last2 = Kotz | first2 = Samuel
| last3 = Balakrishnan | first3 = Narayanaswamy
| title = Continuous Univariate Distributions, Volume 1
| year = 1994
| publisher = Wiley
| isbn=0-471-58495-9
| pages = 171
}} (this is a special case of the generalized inverse normal distribution treated)</ref>
and the first and higher-order moments do not exist.<ref name=Johnson/>

===Inverse Cauchy distribution===

If ''X'' is a [[Cauchy distribution|Cauchy distributed]] (''μ'', ''σ'') random variable, then 1 / X is a Cauchy ( ''μ'' / ''C'', ''σ'' / ''C'' ) random variable where ''C'' = ''μ''2 + ''σ''2.

===Inverse F distribution===

If ''X'' is an [[F distribution|''F''(''ν''1, ''ν''2 ) distributed]] random variable then 1 / ''X'' is an ''F''(''ν''2, ''ν''1 ) random variable.

===Other inverse distributions===

Other inverse distributions include the [[inverse-chi-squared distribution]], the [[inverse-gamma distribution]], the [[inverse-Wishart distribution]], and the [[inverse matrix gamma distribution]].

==Applications==

Inverse distributions are widely used as prior distributions in Bayesian inference for scale parameters.

==See also==

*[[Harmonic mean]]
*[[Ratio distribution]]
*[[Relationships_among_probability_distributions#Reciprocal_of_a_random_variable|Self-reciprocal distributions]]

==References==
{{reflist}}

[[Category:Algebra of random variables]]
[[Category:Types of probability distributions]]

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81.151.118.231 at 01:31, 30 January 2014