Dhistribusi t-student

{{Probability distribution |
  name       =Student’s *t*|
  type       =density|
  pdf_image  =|
  cdf_image  =|
  parameters =*\nu* > 0 degrees of freedom (real)|
  support    =*x* &isin; (−∞; +∞)|
  pdf        =\textstyle\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!|

  cdf        =\begin{matrix}
     \frac{1}{2} + x \Gamma \left( \frac{\nu+1}{2} \right)  \cdot\\[0.5em]
     \frac{\,_2F_1 \left ( \frac{1}{2},\frac{\nu+1}{2};\frac{3}{2};
           -\frac{x^2}{\nu} \right)}
     {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)}
     \end{matrix}where 2*F*1 is the hypergeometric function|
  mean       =0 for \nu > 1, otherwise undefined|
  median     =0|
  mode       =0|
  variance   =\textstyle\frac{\nu}{\nu-2} for \nu > 2, ∞ for 1 \nu ≤ 2, otherwise undefined|
  skewness   =0 for \nu > 3|
  kurtosis   =\textstyle\frac{6}{\nu-4} for \nu > 4|
  entropy    =\begin{matrix}
         \frac{\nu+1}{2}\left[
             \psi \left(\frac{1+\nu}{2} \right)
               - \psi \left(\frac{\nu}{2} \right)
         \right] \\[0.5em]
+ \log{\left[\sqrt{\nu}B \left(\frac{\nu}{2},\frac{1}{2} \right)\right]}
\end{matrix}
* *&psi;*: digamma function,
* *B*: beta function|
|
  mgf        = undefined|
  char       =\textstyle\frac{K_{\nu/2} \left(\sqrt{\nu}|t|)(\sqrt{\nu}|t| \right)^{\nu/2}}{\Gamma(\nu/2)2^{\nu/2-1}} for *\nu > 0
* *K_{\nu}*(*x*): Bessel function
}}
Sajeroning probabilitas lan statistika, **Distribusi t-student** utawa *'Student’s *t*-distribution** (asring dicekak dadi ***t*-distribution'*) iku sawijining distribusi probabilitas lumintu (*continuous probability distribution*  sing dianggo nalika nganakaké èstrimasi aji rata-rata (*mean*) saka sawijining populasi sing ukuran sampelé cilik lan standard déviasi ora diweruhi. 

## Dhéfinisi
### Fungsi dènsiti probabilitas 
Fungsi dhènsitas probabilitas saka distribusi t-Student sing standard ya iku:

:f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}},\!

ing ngendi \nu minangka *drajad kabébasan* lan \Gamma minangka  fungsi Gamma. Bisa uga ditulis:

:f(t) = \frac{1}{\sqrt{\nu}\, B \left (\frac{1}{2}, \frac{\nu}{2}\right )} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!,

ing ngendi *B* iku wujud fungsi Beta.

Kanggo \nu genep,

: \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\,}. 

Kanggo \nu ganjil,

: \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\,}.\!

Gambar-gambar iki nuduhaké dhènsitas saka *t*-distribution tumrap aji \nu sing tansaya mundhak.  Dhistribusi normal dituduhaké kanthi garis biru minangka pembandhing.  Pirsanana yèn *t*-distribution (garis abang) dadi luwih cedhak marang dhistribusi normal nalika aji \nu tansaya gedhé.

## Rujukan 
## Pranala njaba 
*Earliest Known Uses of Some of the Words of Mathematics (S) *(Remarks on the history of the term "Student's distribution")''