View source for Fisher z-distribution

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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|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|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====
<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====
The dispersion proportion is also called the variance ratio.