^Fisher, R. A. (1924). (PDF). Proceedings of the International Congress of Mathematics, Toronto. 2: 805–813. Archived from the original (PDF) on April 12, 2011.
^ abLeo A. Aroian (December 1941). "A study of R. A. Fisher's z distribution and the related F distribution". The Annals of Mathematical Statistics. 12 (4): 429–448. doi:10.1214/aoms/1177731681. JSTOR 2235955.
fisher, distribution, confused, with, fisher, transformation, statistical, distribution, half, logarithm, distribution, variate, fisher, zprobability, density, functionparametersd, displaystyle, freedomsupportx, displaystyle, infty, infty, pdf2, displaystyle, . Not to be confused with Fisher z transformation Fisher s z distribution is the statistical distribution of half the logarithm of an F distribution variate Fisher s zProbability density functionParametersd 1 gt 0 d 2 gt 0 displaystyle d 1 gt 0 d 2 gt 0 deg of freedomSupportx displaystyle x in infty infty PDF2 d 1 d 1 2 d 2 d 2 2 B d 1 2 d 2 2 e d 1 x d 1 e 2 x d 2 d 1 d 2 2 displaystyle frac 2d 1 d 1 2 d 2 d 2 2 B d 1 2 d 2 2 frac e d 1 x left d 1 e 2x d 2 right left d 1 d 2 right 2 Mode0 displaystyle 0 Ronald Fisher z 1 2 log F displaystyle z frac 1 2 log F It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto 1 Nowadays one usually uses the F distribution instead The probability density function and cumulative distribution function can be found by using the F distribution at the value of x e 2 x displaystyle x e 2x However the mean and variance do not follow the same transformation The probability density function is 2 3 f x d 1 d 2 2 d 1 d 1 2 d 2 d 2 2 B d 1 2 d 2 2 e d 1 x d 1 e 2 x d 2 d 1 d 2 2 displaystyle f x d 1 d 2 frac 2d 1 d 1 2 d 2 d 2 2 B d 1 2 d 2 2 frac e d 1 x left d 1 e 2x d 2 right d 1 d 2 2 where B is the beta function When the degrees of freedom becomes large d 1 d 2 displaystyle d 1 d 2 rightarrow infty the distribution approaches normality with mean 2 x 1 2 1 d 2 1 d 1 displaystyle bar x frac 1 2 left frac 1 d 2 frac 1 d 1 right and variance s x 2 1 2 1 d 1 1 d 2 displaystyle sigma x 2 frac 1 2 left frac 1 d 1 frac 1 d 2 right Related distribution EditIf X FisherZ n m displaystyle X sim operatorname FisherZ n m then e 2 X F n m displaystyle e 2X sim operatorname F n m F distribution If X F n m displaystyle X sim operatorname F n m then log X 2 FisherZ n m displaystyle tfrac log X 2 sim operatorname FisherZ n m References Edit Fisher R A 1924 On a Distribution Yielding the Error Functions of Several Well Known Statistics PDF Proceedings of the International Congress of Mathematics Toronto 2 805 813 Archived from the original PDF on April 12 2011 a b Leo A Aroian December 1941 A study of R A Fisher s z distribution and the related F distribution The Annals of Mathematical Statistics 12 4 429 448 doi 10 1214 aoms 1177731681 JSTOR 2235955 Charles Ernest Weatherburn 1961 A first course in mathematical statistics External links EditMathWorld entry Retrieved from https en wikipedia org w index php title Fisher 27s z distribution amp oldid 1086618812, wikipedia, wiki, book, books, library,