# Dhistribusi t-student

Parameters Probability density function Cumulative distribution function $\nu$ > 0 degrees of freedom (real) x ∈ (−∞; +∞) $\textstyle\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!$ $\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{\nu+1}{2} \right) \cdot\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{\nu+1}{2};\frac{3}{2}; -\frac{x^2}{\nu} \right)} {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)} \end{matrix}$ where 2F1 is the hypergeometric function 0 for $\nu$ > 1, otherwise undefined 0 0 $\textstyle\frac{\nu}{\nu-2}$ for $\nu$ > 2, ∞ for 1 < $\nu$ ≤ 2, otherwise undefined 0 for $\nu$ > 3 $\textstyle\frac{6}{\nu-4}$ for $\nu$ > 4 $\begin{matrix} \frac{\nu+1}{2}\left[ \psi \left(\frac{1+\nu}{2} \right) - \psi \left(\frac{\nu}{2} \right) \right] \\[0.5em] + \log{\left[\sqrt{\nu}B \left(\frac{\nu}{2},\frac{1}{2} \right)\right]} \end{matrix}$ undefined $\textstyle\frac{K_{\nu/2} \left(\sqrt{\nu}|t|)(\sqrt{\nu}|t| \right)^{\nu/2}}{\Gamma(\nu/2)2^{\nu/2-1}}$ for $\nu$ > 0 $K_{\nu}$(x): Bessel function[1]

Jroning 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 arupa 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.

 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