查看“︁費雪z分佈”︁的源代码

{{Distinguish|費雪轉換}}
{{Probability distribution
 | name       = Fisher's ''z''
 | type       = density
 | pdf_image  = [[Image:FisherZDistriPDF.svg|325px]]
 | parameters = <math>d_1>0,\ d_2>0</math> deg. of freedom
 | support    = <math>x \in (-\infty; +\infty)\!</math>
 | pdf        = <math>\frac{2d_1^{d_1/2}d_2^{d_2/2}}{B(d_1/2,d_2/2)}\frac{e^{d_1x}}{\left(d_1e^{2x}+d_2\right)^{\left(d_1+d_2\right)/2}}\!</math>
 | mode       = <math>0</math>
}}
[[File:Youngronaldfisher2.JPG|thumb|right|200px|Ronald Fisher]]
'''費雪z分佈'''是[[F分布]]隨機變數的[[自然對數]]0.5倍拉伸量的[[機率分布]]：

: <math>z = \frac 1 2 \ln F </math>

首次由[[羅納德·愛爾默·費雪]]於1924年在[[多倫多]]舉辦的[[國際數學家大會]]的投稿文章所描述。<ref>{{cite journal|last=Fisher |first=R. A. |date=1924 |title=On a Distribution Yielding the Error Functions of Several Well Known Statistics |journal=Proceedings of the International Congress of Mathematics, Toronto |volume=2 |pages=805–813 |url=http://digital.library.adelaide.edu.au/coll/special/fisher/36.pdf |url-status=dead |archive-url=https://web.archive.org/web/20110412083610/http://digital.library.adelaide.edu.au/coll/special//fisher/36.pdf |archive-date=April 12, 2011 }}</ref>現今則經常以[[F分布]]取代之。

費雪z分佈的[[機率密度函數]]與[[累積分布函數]]可由F分布於<math>x' = e^{2x}</math>求得。然而，其平均數與變異數並不能以相同轉換方式求得。

該[[機率密度函數]]為<ref name=Aroian>{{Cite journal
 | author = Leo A. Aroian
 | title = A study of R. A. Fisher's ''z'' distribution and the related F distribution
 | url = https://archive.org/details/sim_annals-of-mathematical-statistics_1941-12_12_4/page/429
 | journal = The Annals of Mathematical Statistics
 | volume = 12
 | issue = 4
 |date=December 1941
 | jstor = 2235955
 | doi = 10.1214/aoms/1177731681
 | pages=429–448
| doi-access = free
 }}</ref><ref>{{Cite book
 | author = Charles Ernest Weatherburn
 | title = A first course in mathematical statistics
| year = 1961
 | url = https://archive.org/details/firstcourseinmat029137mbp
 }}</ref>
: <math>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^{2 x} + d_2\right)^{(d_1+d_2)/2}}</math>
其中''B''為[[Β函數]]。

當[[自由度]]很大（<math>d_1, d_2 \rightarrow \infty</math>）時，該分布逼近[[期望值]]為
<math>\bar{x} = \frac 1 2 \left( \frac 1 {d_2} - \frac 1 {d_1} \right)</math>
且變異數為
<math>\sigma^2_x = \frac 1 2 \left( \frac 1 {d_1} + \frac 1 {d_2} \right)</math>
的[[常態分布]]。<ref name=Aroian/>

==相關分布==
*若<math> X \sim \operatorname{FisherZ}(n,m) </math>則<math>e^{2X} \sim \operatorname{F}(n,m)</math>（[[F分布]]）。
*若<math> X \sim \operatorname{F}(n,m) </math>則<math>\tfrac{\log X}{2} \sim \operatorname{FisherZ}(n,m)</math>。




== 參考資料 ==
{{Reflist|2}}

== 外部連結 ==
* [http://mathworld.wolfram.com/Fishersz-Distribution.html MathWorld entry] {{Wayback|url=http://mathworld.wolfram.com/Fishersz-Distribution.html |date=20210429040520 }}
{{概率分布}}
[[Category:连续分布]]