Sajeroning probabilitas lan statistika , Distribusi t-student utawa Student’s t -distribution (asring dicekak dadi t -distributioncontinuous probability distribution sing dianggo nalika nganakaké èstrimasi aji rata-rata (mean ) saka sawijining populasi sing ukuran sampel é cilik lan standard déviasi ora diweruhi.
Student’s t
Probability density function Cumulative distribution function Parameters
ν {\displaystyle \nu } degrees of freedom (real )
Support
x ∈ (−∞; +∞)
PDF
Γ ( ν + 1 2 ) ν π Γ ( ν 2 ) ( 1 + x 2 ν ) − ν + 1 2 {\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
1 2 + x Γ ( ν + 1 2 ) ⋅ 2 F 1 ( 1 2 , ν + 1 2 ; 3 2 ; − x 2 ν ) π ν Γ ( ν 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}}} 2 F 1 is the hypergeometric function
Mean
0 for ν {\displaystyle \nu } Median
0
Mode
0
Variance
ν ν − 2 {\displaystyle \textstyle {\frac {\nu }{\nu -2}}} ν {\displaystyle \nu } ν {\displaystyle \nu } Skewness
0 for ν {\displaystyle \nu } Ex. kurtosis
6 ν − 4 {\displaystyle \textstyle {\frac {6}{\nu -4}}} ν {\displaystyle \nu } Entropy
ν + 1 2 [ ψ ( 1 + ν 2 ) − ψ ( ν 2 ) ] + log [ ν B ( ν 2 , 1 2 ) ] {\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}}} MGF
undefined
CF
K ν / 2 ( ν | t | ) ( ν | t | ) ν / 2 Γ ( ν / 2 ) 2 ν / 2 − 1 {\displaystyle \textstyle {\frac {K_{\nu /2}\left({\sqrt {\nu }}|t|)({\sqrt {\nu }}|t|\right)^{\nu /2}}{\Gamma (\nu /2)2^{\nu /2-1}}}} ν {\displaystyle \nu }
Fungsi dènsiti probabilitas
besut
Fungsi dhènsitas probabilitas saka distribusi t-Student sing standard ya iku:
f ( t ) = Γ ( ν + 1 2 ) ν π Γ ( ν 2 ) ( 1 + t 2 ν ) − ν + 1 2 , {\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 ν {\displaystyle \nu } drajad kabébasan Γ {\displaystyle \Gamma } fungsi Gamma . Bisa uga ditulis:
f ( t ) = 1 ν B ( 1 2 , ν 2 ) ( 1 + t 2 ν ) − ν + 1 2 , {\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 ν {\displaystyle \nu }
Γ ( ν + 1 2 ) ν π Γ ( ν 2 ) = ( ν − 1 ) ( ν − 3 ) ⋯ 5 ⋅ 3 2 ν ( ν − 2 ) ( ν − 4 ) ⋯ 4 ⋅ 2 . {\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 ν {\displaystyle \nu }
Γ ( ν + 1 2 ) ν π Γ ( ν 2 ) = ( ν − 1 ) ( ν − 3 ) ⋯ 4 ⋅ 2 π ν ( ν − 2 ) ( ν − 4 ) ⋯ 5 ⋅ 3 . {\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 ν {\displaystyle \nu } t -distribution (garis abang) dadi luwih cedhak marang dhistribusi normal nalika aji ν {\displaystyle \nu }
Density of the t -distribution (red) for 1, 2, 3, 5, 10, and 30 df compared to the standard normal distribution (blue).
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
besut
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02732e546784af1fb16d0dc1bb65dd743e2284ad)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d02573057e45a7d497ab31ced38dbd9a5901f77)
