# Fisher z-distribution

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