# 司徒頓t分布

（重定向自学生t-分布

参数 概率密度函數 累積分布函數 ${\displaystyle \nu >0\!}$自由度 ${\displaystyle x\in (-\infty ;+\infty )\!}$ ${\displaystyle {\frac {\Gamma ((\nu +1)/2)}{{\sqrt {\nu \pi }}\,\Gamma (\nu /2)\,(1+x^{2}/\nu )^{(\nu +1)/2}}}\!}$ ${\displaystyle {\frac {1}{2}}+{\frac {x\Gamma \left((\nu +1)/2\right)\,_{2}F_{1}\left({\frac {1}{2}},(\nu +1)/2;{\frac {3}{2}};-{\frac {x^{2}}{\nu }}\right)}{{\sqrt {\pi \nu }}\,\Gamma (\nu /2)}}}$其中：${\displaystyle \,_{2}F_{1}}$是超几何函数 ${\displaystyle \nu >1}$时为${\displaystyle 0}$，${\displaystyle \nu =1}$时未定义 ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle \nu >2}$时为${\displaystyle {\frac {\nu }{\nu -2}}\!}$，否则为无穷大 ${\displaystyle \nu >3}$时为${\displaystyle 0}$ ${\displaystyle \nu >4}$时为${\displaystyle {\frac {6}{\nu -4}}\!}$ ${\displaystyle {\begin{matrix}{\frac {\nu +1}{2}}\left[\psi ({\frac {1+\nu }{2}})-\psi ({\frac {\nu }{2}})\right]\\[0.5em]+\log {\left[{\sqrt {\nu }}B({\frac {\nu }{2}},{\frac {1}{2}})\right]}\end{matrix}}}$ ${\displaystyle \psi }$: 双Γ函数, ${\displaystyle B}$: 贝塔函数 未定义 ${\displaystyle {\frac {K_{\nu /2}({\sqrt {\nu }}|t|)({\sqrt {\nu }}|t|)^{\nu /2}}{\Gamma (\nu /2)2^{\nu /2-1}}},\;\nu >0}$ ${\displaystyle K_{\nu }(x)}$: 第二类修正貝塞爾函數

t 分布的推导最早由德國大地测量学家于1876年提出，并由德國数学家雅各布·魯洛斯英语Jacob Lüroth证明。[1][2]

## 描述

${\displaystyle {\overline {X}}_{n}={\frac {X_{1}+\cdots +X_{n}}{n}}}$

${\displaystyle {S_{n}}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}\left(X_{i}-{\overline {X}}_{n}\right)^{2}}$

${\displaystyle Z={\frac {X-\mu }{\frac {\sigma }{\sqrt {n}}}}}$

${\displaystyle T={\frac {X-\mu }{\frac {S_{n}}{\sqrt {n}}}}}$

T機率密度函數是：

${\displaystyle f(t)={\frac {\Gamma ({\frac {\nu +1}{2}})}{{\sqrt {\nu \pi \,}}\,\Gamma ({\frac {\nu }{2}})}}(1+{\frac {t^{2}}{\nu }})^{\frac {-(\nu +1)}{2}}}$

${\displaystyle \nu }$  等于n − 1。 T的分布称为t 分布母數${\displaystyle \nu }$  一般被称为自由度

${\displaystyle \Gamma }$  伽玛函数。 如果${\displaystyle \nu }$ 是偶数,

${\displaystyle {\frac {\Gamma ({\frac {\nu +1}{2}})}{{\sqrt {\nu \pi }}\,\Gamma ({\frac {\nu }{2}})}}={\frac {(\nu -1)(\nu -3)\cdots 5\cdot 3}{2{\sqrt {\nu }}(\nu -2)(\nu -4)\cdots 4\cdot 2\,}}\cdot }$

${\displaystyle {\frac {\Gamma ({\frac {\nu +1}{2}})}{{\sqrt {\nu \pi }}\,\Gamma ({\frac {\nu }{2}})}}={\frac {(\nu -1)(\nu -3)\cdots 4\cdot 2}{\pi {\sqrt {\nu }}(\nu -2)(\nu -4)\cdots 5\cdot 3\,}}\cdot \!}$

T機率密度函數的形状类似于期望值为0方差为1的正态分布，但更低更宽。随着自由度${\displaystyle \nu }$ 的增加，则越来越接近期望值为0方差为1的正态分布。

 1 degree of freedom 2 degrees of freedom 3 degrees of freedom 5 degrees of freedom 10 degrees of freedom 30 degrees of freedom

T分布的概率累计函数，用不完全贝塔函数I表示：

${\displaystyle F(t)=\int _{-\infty }^{t}f(u)\,du=1-{\tfrac {1}{2}}I_{x(t)}\left({\tfrac {\nu }{2}},{\tfrac {1}{2}}\right),}$

${\displaystyle x(t)={\frac {\nu }{t^{2}+\nu }}.}$

T分布的矩为：

${\displaystyle E(T^{k})={\begin{cases}0&{\mbox{k odd}},0

## 学生t 分布置信区间的推导

${\displaystyle \Pr(-A

${\displaystyle \Pr(T 是相同的

A是这个概率分布的第95个百分点

${\displaystyle \Pr \left(-A<{{\overline {X}}_{n}-\mu \over S_{n}/{\sqrt {n}}}

${\displaystyle \Pr \left({\overline {X}}_{n}-A{S_{n} \over {\sqrt {n}}}<\mu <{\overline {X}}_{n}+A{S_{n} \over {\sqrt {n}}}\right)=0.9}$

${\displaystyle {\overline {X}}_{n}\pm A{\frac {S_{n}}{\sqrt {n}}}}$

## 计算

Pr(T < −2.132) = 1 − Pr(T > −2.132) = 1 − 0.95 = 0.05,

Pr(−2.132 < T < 2.132) = 1 − 2(0.05) = 0.9.

1 1.000 1.376 1.963 3.078 6.314 12.71 31.82 63.66 127.3 318.3 636.6
2 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 14.09 22.33 31.60
3 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 7.453 10.21 12.92
4 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610
5 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 4.773 5.893 6.869
6 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.959
7 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408
8 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.041
9 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781
10 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587
11 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.437
12 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.318
13 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.221
14 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140
15 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.073
16 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4.015
17 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3.965
18 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.922
19 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.883
20 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.850
21 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3.819
22 0.686 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3.792
23 0.685 0.858 1.060 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.767
24 0.685 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.091 3.467 3.745
25 0.684 0.856 1.058 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3.725
26 0.684 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.707
27 0.684 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.690
28 0.683 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3.674
29 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.659
30 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3.646
40 0.681 0.851 1.050 1.303 1.684 2.021 2.423 2.704 2.971 3.307 3.551
50 0.679 0.849 1.047 1.299 1.676 2.009 2.403 2.678 2.937 3.261 3.496
60 0.679 0.848 1.045 1.296 1.671 2.000 2.390 2.660 2.915 3.232 3.460
80 0.678 0.846 1.043 1.292 1.664 1.990 2.374 2.639 2.887 3.195 3.416
100 0.677 0.845 1.042 1.290 1.660 1.984 2.364 2.626 2.871 3.174 3.390
120 0.677 0.845 1.041 1.289 1.658 1.980 2.358 2.617 2.860 3.160 3.373
${\displaystyle \infty }$  0.674 0.842 1.036 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291

## 範例

${\displaystyle {\overline {X}}_{n}\pm A{\frac {S_{n}}{\sqrt {n}}}}$

${\displaystyle 10+1.37218{\frac {\sqrt {2}}{\sqrt {11}}}=10.58510.}$

${\displaystyle 10-1.37218{\frac {\sqrt {2}}{\sqrt {11}}}=9.41490.}$

${\displaystyle 10\pm 1.37218{\frac {\sqrt {2}}{\sqrt {11}}}=[9.41490,10.58510]}$

## 參考文獻

1. ^ Pfanzagl, J.; Sheynin, O. A forerunner of the t-distribution (Studies in the history of probability and statistics XLIV). Biometrika. 1996, 83 (4): 891–898. MR 1766040. doi:10.1093/biomet/83.4.891.
2. ^ Sheynin, O. Helmert’s work in the theory of errors. Arch. Hist. Exact Sci. 1995, 49: 73–104. doi:10.1007/BF00374700.
3. ^ Moore, David S. Introduction to the Practice of SATISTICS. George P. McCabe, Bruce A. Craig 7th International Edition. New York: W. H. Freeman and Company. 2012: p. 401. ISBN 978-1-4292-8664-0 （英语）.