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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.
 
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.
  
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" />.
+
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" />.
  
 
====References====
 
====References====

Revision as of 20:15, 2 April 2018

A continuous probability distribution on the real line with density

The parameters are called the degrees of freedom. The characteristic function has the form

The mathematical expectation and the variance are equal to and , respectively.

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

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