Dhistribusi t-student

Cithakan:Probability distribution 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:

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

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

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

ing ngendi B iku wujud fungsi Beta.

Kanggo $\nu$ genep,

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

Kanggo $\nu$ ganjil,

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

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.

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.

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

Pranala njaba

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