Dhistribusi t-student

Student’s t

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \nu }$ > 0 degrees of freedom (real)
Support	x ∈ (−∞; +∞)
PDF	${\displaystyle \textstyle {\frac {\Gamma \left({\frac {\nu +1}{2}\right)}{\sqrt {\nu \pi }\,\Gamma \left({\frac {\nu }{2}\right)}\left(1+{\frac {x^{2}{\nu }\right)^{-{\frac {\nu +1}{2}\!}$
CDF	${\displaystyle {\begin{matrix}{\frac {1}{2}+x\Gamma \left({\frac {\nu +1}{2}\right)\cdot \\[0.5em]{\frac {\,_{2}F_{1}\left({\frac {1}{2},{\frac {\nu +1}{2};{\frac {3}{2};-{\frac {x^{2}{\nu }\right)}{\sqrt {\pi \nu }\,\Gamma \left({\frac {\nu }{2}\right)}\end{matrix}$ where 2F1 is the hypergeometric function
Mean	0 for ${\displaystyle \nu }$ > 1, otherwise undefined
Median	0
Mode	0
Variance	${\displaystyle \textstyle {\frac {\nu }{\nu -2}$ for ${\displaystyle \nu }$ > 2, ∞ for 1 < ${\displaystyle \nu }$ ≤ 2, otherwise undefined
Skewness	0 for ${\displaystyle \nu }$ > 3
Ex. kurtosis	${\displaystyle \textstyle {\frac {6}{\nu -4}$ for ${\displaystyle \nu }$ > 4
Entropy	${\displaystyle {\begin{matrix}{\frac {\nu +1}{2}\left[\psi \left({\frac {1+\nu }{2}\right)-\psi \left({\frac {\nu }{2}\right)\right]\\[0.5em]+\log {\left[{\sqrt {\nu }B\left({\frac {\nu }{2},{\frac {1}{2}\right)\right]}\end{matrix}$ ψ: digamma function, B: beta function
MGF	undefined
CF	${\displaystyle \textstyle {\frac {K_{\nu /2}\left({\sqrt {\nu }|t|)({\sqrt {\nu }|t|\right)^{\nu /2}{\Gamma (\nu /2)2^{\nu /2-1}$ 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.

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.

Density of the t-distribution (red) for 1, 2, 3, 5, 10, and 30 df compared to the standard normal distribution (blue).
Previous plots shown in green.

• Earliest Known Uses of Some of the Words of Mathematics (S) (Remarks on the history of the term "Student's distribution")