Dhistribusi t-student

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Student’s t
Probability density function
Student t pdf.svg
Cumulative distribution function
Student t cdf.svg
Parameters \nu > 0 degrees of freedom (real)
Support x ∈ (−∞; +∞)
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 2F1 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
Ex. 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}
MGF undefined
CF \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

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[sunting | sunting sumber]

Fungsi dènsiti probabilitas[sunting | sunting sumber]

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.

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

Rujukan[sunting | sunting sumber]

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

Pranala njaba[sunting | sunting sumber]