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A continuous probability distribution on the real line with density
 
A continuous probability distribution on the real line with density
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f0405002.png" /></td> </tr></table>
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$$
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f ( x) =
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$$
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$$
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= \
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2m _ {1} ^ {m _ {1} /2 } m _ {2} ^ {m _ {2} /2 }
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\frac{\Gamma ( ( m _ {1} + m _ {2} )/2) e ^ {m _ {1} x
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} }{\Gamma ( {m _ {1} /2 } ) \Gamma ( {m _ {2} /2 } ) ( m _ {1} e  ^ {2x} + m _ {2} )
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^ {( m _ {1} + m _ {2} )/2 }} .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f0405003.png" /></td> </tr></table>
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The parameters  $  m _ {1} , m _ {2} \geq  1 $
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are called the degrees of freedom. The characteristic function has the form
  
The parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f0405004.png" /> are called the degrees of freedom. The characteristic function has the form
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$$
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\phi ( t)  = \
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\left (
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\frac{m _ {2} }{m _ {1} }
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\right ) ^ { {{it } /2 } }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f0405005.png" /></td> </tr></table>
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\frac{\Gamma ( {( m _ {1} + it)/2 } ) \Gamma ( {( m _ {2} - it)/2 } ) }{\Gamma ( { {m _ {1} } /2 } ) \Gamma ( { {m _ {2} } /2 } ) }
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.
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$$
  
The mathematical expectation and the variance are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f0405006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f0405007.png" />, respectively.
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The mathematical expectation and the variance are equal to $  ( 1/m _ {1} - 1/m _ {2} )/2 $
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and $  ( 1/m _ {1} + 1/m _ {2} )/2 $,  
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respectively.
  
If the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f0405008.png" /> has the [[Fisher-F-distribution|Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f0405009.png" />-distribution]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050011.png" /> degrees of freedom, then the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050012.png" /> has the Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050013.png" />-distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050015.png" /> degrees of freedom. Along with the Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050016.png" />-distribution, known as the distribution of the [[Dispersion proportion|dispersion proportion]], the Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050017.png" />-distribution was originally introduced in the [[analysis of variance]] by R.A. Fisher (1924). His intention was that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050018.png" />-distribution should be the basic distribution for testing statistical hypotheses in the analysis of variance. The Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050019.png" />-distribution was tabulated at the same time, and the first research was concerned with the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050020.png" />, although in modern mathematical statistics one uses the simpler statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040500/f04050021.png" />.
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If the random variable $  F $
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has the [[Fisher-F-distribution|Fisher $  F $-distribution]] with $  m _ {1} $
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and $  m _ {2} $
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degrees of freedom, then the quantity $  z = (  \mathop{\rm log}  F)/2 $
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has the Fisher $  z $-distribution with $  m _ {1} $
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and $  m _ {2} $
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degrees of freedom. Along with the Fisher $  F $-distribution, known as the distribution of the [[Dispersion proportion|dispersion proportion]], the Fisher $  z $-distribution was originally introduced in the [[analysis of variance]] by R.A. Fisher (1924). His intention was that the $  z $-distribution should be the basic distribution for testing statistical hypotheses in the analysis of variance. The Fisher $  z $-distribution was tabulated at the same time, and the first research was concerned with the statistic $  z $,  
 +
although in modern mathematical statistics one uses the simpler statistic $  F $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.A. Fisher,  "On a distribution yielding the error functions of several well-known statistics" , ''Proc. Internat. Congress mathematicians (Toronto 1924)'' , '''2''' , Univ. Toronto Press  (1928)  pp. 805–813</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.A. Fisher,  "On a distribution yielding the error functions of several well-known statistics" , ''Proc. Internat. Congress mathematicians (Toronto 1924)'' , '''2''' , Univ. Toronto Press  (1928)  pp. 805–813</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The dispersion proportion is also called the variance ratio.
 
The dispersion proportion is also called the variance ratio.

Latest revision as of 04:59, 24 February 2022


A continuous probability distribution on the real line with density

$$ f ( x) = $$

$$ = \ 2m _ {1} ^ {m _ {1} /2 } m _ {2} ^ {m _ {2} /2 } \frac{\Gamma ( ( m _ {1} + m _ {2} )/2) e ^ {m _ {1} x } }{\Gamma ( {m _ {1} /2 } ) \Gamma ( {m _ {2} /2 } ) ( m _ {1} e ^ {2x} + m _ {2} ) ^ {( m _ {1} + m _ {2} )/2 }} . $$

The parameters $ m _ {1} , m _ {2} \geq 1 $ are called the degrees of freedom. The characteristic function has the form

$$ \phi ( t) = \ \left ( \frac{m _ {2} }{m _ {1} } \right ) ^ { {{it } /2 } } \frac{\Gamma ( {( m _ {1} + it)/2 } ) \Gamma ( {( m _ {2} - it)/2 } ) }{\Gamma ( { {m _ {1} } /2 } ) \Gamma ( { {m _ {2} } /2 } ) } . $$

The mathematical expectation and the variance are equal to $ ( 1/m _ {1} - 1/m _ {2} )/2 $ and $ ( 1/m _ {1} + 1/m _ {2} )/2 $, respectively.

If the random variable $ F $ has the Fisher $ F $-distribution with $ m _ {1} $ and $ m _ {2} $ degrees of freedom, then the quantity $ z = ( \mathop{\rm log} F)/2 $ has the Fisher $ z $-distribution with $ m _ {1} $ and $ m _ {2} $ degrees of freedom. Along with the Fisher $ F $-distribution, known as the distribution of the dispersion proportion, the Fisher $ z $-distribution was originally introduced in the analysis of variance by R.A. Fisher (1924). His intention was that the $ z $-distribution should be the basic distribution for testing statistical hypotheses in the analysis of variance. The Fisher $ z $-distribution was tabulated at the same time, and the first research was concerned with the statistic $ z $, although in modern mathematical statistics one uses the simpler statistic $ F $.

References

[1] R.A. Fisher, "On a distribution yielding the error functions of several well-known statistics" , Proc. Internat. Congress mathematicians (Toronto 1924) , 2 , Univ. Toronto Press (1928) pp. 805–813

Comments

The dispersion proportion is also called the variance ratio.

How to Cite This Entry:
Fisher z-distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fisher_z-distribution&oldid=43081
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article