Studentova t-porazdelitev (tudi t-porazdelitev ali Študentova t-porazdelitev ) je zvezna verjetnostna porazdelitev .
Student's t
parametri
ν
>
0
{\displaystyle \nu >0}
prostostne stopnje (realno število ) Interval
x
∈
(
−
∞
,
∞
)
{\displaystyle x\in (-\infty ,\infty )}
gostota verjetnosti (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}}}\!}
zbirna funkcija verjetnosti (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)\times \\[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 2 F 1 is the hipergeometrična funkcija pričakovana vrednost
0 for
ν
>
1
{\displaystyle \nu >1}
mediana
0 modus
0 varianca
ν
ν
−
2
{\displaystyle \textstyle {\frac {\nu }{\nu -2}}}
ν
>
2
{\displaystyle \nu >2}
1
<
ν
≤
2
{\displaystyle 1<\nu \leq 2}
nesimetričnost
0 for
ν
>
3
{\displaystyle \nu >3}
sploščenost
6
ν
−
4
{\displaystyle \textstyle {\frac {6}{\nu -4}}}
ν
>
4
{\displaystyle \nu >4}
2
<
ν
≤
4
{\displaystyle 2<\nu \leq 4}
entropija
ν
+
1
2
[
ψ
(
1
+
ν
2
)
−
ψ
(
ν
2
)
]
+
ln
[
ν
B
(
ν
2
,
1
2
)
]
(nats)
{\displaystyle {\begin{matrix}{\frac {\nu +1}{2}}\left[\psi \left({\frac {1+\nu }{2}}\right)-\psi \left({\frac {\nu }{2}}\right)\right]\\[0.5em]+\ln {\left[{\sqrt {\nu }}B\left({\frac {\nu }{2}},{\frac {1}{2}}\right)\right]}\,{\scriptstyle {\text{(nats)}}}\end{matrix}}}
funkcija generiranja momentov (mgf)
ni definirana karakteristična funkcija
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|\right)\cdot \left({\sqrt {\nu }}|t|\right)^{\nu /2}}{\Gamma (\nu /2)2^{\nu /2-1}}}}
ν
>
0
{\displaystyle \nu >0}
Studentovo t-porazdelitev je odkril William Sealy Gosset (1876–1937) v letu 1908. Njeno odkritje je objavil pod psevdonimom Student (študent). Gosset je bil pivovar v pivovarni pri Guinnessu . Porazdelitev je odkril med raziskavo vpliva kvasovk na kakovost piva . Pozneje je ameriški statistik in ekonomski teoretik Harold Hotelling (1895 – 1973) razvil t porazdelitev. Ime porazdelitve pa je ostalo.
Studentova t-porazdelitev je verjetnostna porazdelitev razmerja
Z
V
/
ν
=
Z
ν
/
V
{\displaystyle {\frac {Z}{\sqrt {V/\nu \ }}}=Z{\sqrt {\nu /V}}}
kjer ima
Lastnosti t porazdelitve
uredi
Funkcija gostote verjetnosti
uredi
Funkcija gostote verjetnosti za t porazdelitev je
Γ
(
ν
+
1
2
)
ν
π
Γ
(
ν
2
)
(
1
+
x
2
ν
)
−
(
ν
+
1
2
)
{\displaystyle {\frac {\Gamma ({\frac {\nu +1}{2}})}{{\sqrt {\nu \pi }}\,\Gamma ({\frac {\nu }{2}})}}\left(1+{\frac {x^{2}}{\nu }}\right)^{-({\frac {\nu +1}{2}})}\!}
kjer je
Γ
(
z
)
{\displaystyle \Gamma (z)\!}
funkcija gama .
ν
{\displaystyle \nu }
Kadar je
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)(3)}{2{\sqrt {\nu }}(\nu -2)(\nu -4)\cdots (4)(2)\,}}.}
Kadar pa je
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)(2)}{\pi {\sqrt {\nu }}(\nu -2)(\nu -4)\cdots (5)(3)\,}}.\!}
Zbirna funkcija verjetnosti
uredi
Zbirna funkcija verjetnosti je enaka
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 ({\frac {\nu }{2}})}}\end{matrix}}}
kjer je
Zbirno funkcijo verjetnosti pa lahko izrazimo tudi s pomočjo nepopolne funkcije beta:
∫
−
∞
t
f
(
u
)
d
u
=
I
x
(
ν
2
,
ν
2
)
=
B
(
x
;
ν
2
,
ν
2
)
B
(
ν
2
,
ν
2
)
{\displaystyle \int _{-\infty }^{t}f(u)\,du=I_{x}\left({\frac {\nu }{2}},{\frac {\nu }{2}}\right)={\frac {B\left(x;{\frac {\nu }{2}},{\frac {\nu }{2}}\right)}{B\left({\frac {\nu }{2}},{\frac {\nu }{2}}\right)}}}
kjer je
x
=
t
+
t
2
+
ν
2
t
2
+
ν
.
{\displaystyle x={\frac {t+{\sqrt {t^{2}+\nu }}}{2{\sqrt {t^{2}+\nu }}}}.}
B
(
x
,
a
,
b
)
{\displaystyle B(x,a,b)\!}
funkcija beta
uredi
Pričakovana vrednost je enaka
0
za
ν
>
1
{\displaystyle 0{\text{ za }}\nu >1\!}
drugje je nedefinirana.
Varianca je enaka
ν
ν
−
2
za vrednosti
ν
>
2
{\displaystyle {\frac {\nu }{\nu -2}}{\text{ za vrednosti }}\nu >2\!}
∞
{\displaystyle \infty \!}
1
<
ν
≤
2
{\displaystyle 1<\nu \leq 2}
Sploščenost je enaka
6
ν
−
4
za
ν
>
4
{\displaystyle {\frac {6}{\nu -4}}{\text{ za }}\nu >4\!}
Funkcija generiranja momentov
uredi
Funkcija generiranja momentov ni določena.
Povezave z drugimi porazdelitvami
uredi
Slučajna spremenljivka
Y
{\displaystyle Y\!}
F porazdelitev
Y
∼
F
(
ν
1
=
1
,
ν
2
=
ν
)
{\displaystyle Y\sim \mathrm {F} (\nu _{1}=1,\nu _{2}=\nu )\!}
Y
=
X
2
{\displaystyle Y=X^{2}\!}
X
{\displaystyle X\!}
X
∼
t
(
ν
)
{\displaystyle X\sim \mathrm {t} (\nu )\!}
Slučajna spremenljivka
Y
{\displaystyle Y\!}
normalno porazdelitev
Y
∼
N
(
0
,
1
)
{\displaystyle Y\sim \mathrm {N} (0,1)\!}
Y
=
lim
ν
→
∞
X
{\displaystyle Y=\lim _{\nu \to \infty }X}
X
{\displaystyle X\!}
X
∼
t
(
ν
)
{\displaystyle X\sim \mathrm {t} (\nu )\!}
Slučajna spremenljivka
X
{\displaystyle X\!}
Cauchyjevo porazdelitev
X
∼
C
a
u
c
h
y
(
0
,
1
)
{\displaystyle X\sim \mathrm {Cauchy} (0,1)\!}
X
{\displaystyle X\!}
X
∼
t
(
ν
=
1
)
{\displaystyle X\sim \mathrm {t} (\nu =1)\!}
![{\displaystyle {\begin{matrix}{\frac {1}{2}}+x\Gamma \left({\frac {\nu +1}{2}}\right)\times \\[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/7c3c84e8f1257dce799724d08e3b08389944045d)
![{\displaystyle {\begin{matrix}{\frac {\nu +1}{2}}\left[\psi \left({\frac {1+\nu }{2}}\right)-\psi \left({\frac {\nu }{2}}\right)\right]\\[0.5em]+\ln {\left[{\sqrt {\nu }}B\left({\frac {\nu }{2}},{\frac {1}{2}}\right)\right]}\,{\scriptstyle {\text{(nats)}}}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e64e6a7fd1bb08a7129701a00f10b4dc673c589)
![{\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 ({\frac {\nu }{2}})}}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0600c994f77081f5b4a378de62d0d1bae0f59188)
