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# Dhistribusi t-student

Parameters Probability density function Cumulative distribution function ${\displaystyle \nu }$ > 0 degrees of freedom (real) x ∈ (−∞; +∞) ${\displaystyle \textstyle {\frac {\Gamma \left({\frac {\nu +1}{2}}\right)}{{\sqrt {\nu \pi }}\,\Gamma \left({\frac {\nu }{2}}\right)}}\left(1+{\frac {x^{2}}{\nu }}\right)^{-{\frac {\nu +1}{2}}}\!}$ ${\displaystyle {\begin{matrix}{\frac {1}{2}}+x\Gamma \left({\frac {\nu +1}{2}}\right)\cdot \\[0.5em]{\frac {\,_{2}F_{1}\left({\frac {1}{2}},{\frac {\nu +1}{2}};{\frac {3}{2}};-{\frac {x^{2}}{\nu }}\right)}{{\sqrt {\pi \nu }}\,\Gamma \left({\frac {\nu }{2}}\right)}}\end{matrix}}}$where 2F1 is the hypergeometric function 0 for ${\displaystyle \nu }$ > 1, otherwise undefined 0 0 ${\displaystyle \textstyle {\frac {\nu }{\nu -2}}}$ for ${\displaystyle \nu }$ > 2, ∞ for 1 < ${\displaystyle \nu }$ ≤ 2, otherwise undefined 0 for ${\displaystyle \nu }$ > 3 ${\displaystyle \textstyle {\frac {6}{\nu -4}}}$ for ${\displaystyle \nu }$ > 4 ${\displaystyle {\begin{matrix}{\frac {\nu +1}{2}}\left[\psi \left({\frac {1+\nu }{2}}\right)-\psi \left({\frac {\nu }{2}}\right)\right]\\[0.5em]+\log {\left[{\sqrt {\nu }}B\left({\frac {\nu }{2}},{\frac {1}{2}}\right)\right]}\end{matrix}}}$ undefined ${\displaystyle \textstyle {\frac {K_{\nu /2}\left({\sqrt {\nu }}|t|)({\sqrt {\nu }}|t|\right)^{\nu /2}}{\Gamma (\nu /2)2^{\nu /2-1}}}}$ for ${\displaystyle \nu }$ > 0 ${\displaystyle K_{\nu }}$(x): Bessel function[1]

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:

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

ing ngendi ${\displaystyle \nu }$ minangka drajad kabébasan lan ${\displaystyle \Gamma }$ minangka fungsi Gamma. Bisa uga ditulis:

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

ing ngendi B iku wujud fungsi Beta.

Kanggo ${\displaystyle \nu }$ genep,

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

Kanggo ${\displaystyle \nu }$ ganjil,

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

Gambar-gambar iki nuduhaké dhènsitas saka t-distribution tumrap aji ${\displaystyle \nu }$ 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 ${\displaystyle \nu }$ tansaya gedhé.

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

## Rujukan

1. Hurst, Simon, The Characteristic Function of the Student-t Distribution, Financial Mathematics Research Report No. FMRR006-95, Statistics Research Report No. SRR044-95