Definition Student's t-distribution

-



1 definition

1.1 probability density function
1.2 cumulative distribution function
1.3 special cases





definition
probability density function

student s t-distribution has probability density function given by







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}}},\!}



where



ν


{\displaystyle \nu }

number of degrees of freedom ,



Γ


{\displaystyle \gamma }

gamma function. may written as







f
(
t
)
=


1

2


ν




b

(


1
2


,


ν
2


)





(
1
+



t

2


ν


)



−



ν
+
1

2





,


{\displaystyle f(t)={\frac {1}{2{\sqrt {\nu }}\,\mathrm {b} ({\frac {1}{2}},{\frac {\nu }{2}})}}\left(1+{\frac {t^{2}}{\nu }}\right)^{\!-{\frac {\nu +1}{2}}}\!,}



where b beta function. in particular integer valued degrees of freedom



ν


{\displaystyle \nu }

have:


for



ν
>
1


{\displaystyle \nu >1}

even,










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\,}}\cdot }



for



ν
>
1


{\displaystyle \nu >1}

odd,










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\,}}\cdot \!}



the probability density function symmetric, , overall shape resembles bell shape of distributed variable mean 0 , variance 1, except bit lower , wider. number of degrees of freedom grows, t-distribution approaches normal distribution mean 0 , variance 1. reason




ν



{\displaystyle {\nu }}

known normality parameter .


the following images show density of t-distribution increasing values of



ν


{\displaystyle \nu }

. normal distribution shown blue line comparison. note t-distribution (red line) becomes closer normal distribution



ν


{\displaystyle \nu }

increases.



cumulative distribution function

the cumulative distribution function can written in terms of i, regularized incomplete beta function. t > 0,







f
(
t
)
=

∫

−
∞


t


f
(
u
)

d
u
=
1
−



1
2




i

x
(
t
)



(



ν
2



,



1
2



)

,


{\displaystyle f(t)=\int _{-\infty }^{t}f(u)\,du=1-{\tfrac {1}{2}}i_{x(t)}\left({\tfrac {\nu }{2}},{\tfrac {1}{2}}\right),}



where







x
(
t
)
=


ν


t

2


+
ν



.


{\displaystyle x(t)={\frac {\nu }{t^{2}+\nu }}.}



other values obtained symmetry. alternative formula, valid




t

2


<
ν


{\displaystyle t^{2}<\nu }

, is








∫

−
∞


t


f
(
u
)

d
u
=



1
2



+
t



Γ

(



1
2



(
ν
+
1
)
)





π
ν



Γ

(



ν
2



)










2



f

1



(



1
2



,



1
2



(
ν
+
1
)
;



3
2



;
−




t

2


ν



)

,


{\displaystyle \int _{-\infty }^{t}f(u)\,du={\tfrac {1}{2}}+t{\frac {\gamma \left({\tfrac {1}{2}}(\nu +1)\right)}{{\sqrt {\pi \nu }}\,\gamma \left({\tfrac {\nu }{2}}\right)}}\,{}_{2}f_{1}\left({\tfrac {1}{2}},{\tfrac {1}{2}}(\nu +1);{\tfrac {3}{2}};-{\tfrac {t^{2}}{\nu }}\right),}



where 2f1 particular case of hypergeometric function.


for information on inverse cumulative distribution function, see quantile function#student s t-distribution.


special cases

certain values of



ν


{\displaystyle \nu }

give simple form.







ν
=
1


{\displaystyle \nu =1}




distribution function:








f
(
t
)
=



1
2



+



1
π



arctan
⁡
(
t
)
.


{\displaystyle f(t)={\tfrac {1}{2}}+{\tfrac {1}{\pi }}\arctan(t).}






density function:








f
(
t
)
=


1

π
(
1
+

t

2


)



.


{\displaystyle f(t)={\frac {1}{\pi (1+t^{2})}}.}






see cauchy distribution






ν
=
2


{\displaystyle \nu =2}




distribution function:








f
(
t
)
=



1
2



+


t

2


2
+

t

2







.


{\displaystyle f(t)={\tfrac {1}{2}}+{\frac {t}{2{\sqrt {2+t^{2}}}}}.}






density function:








f
(
t
)
=


1


(
2
+

t

2


)



3
2





.


{\displaystyle f(t)={\frac {1}{\left(2+t^{2}\right)^{\frac {3}{2}}}}.}










ν
=
3


{\displaystyle \nu =3}




density function:








f
(
t
)
=



6


3




π


(
3
+

t

2


)


2





.


{\displaystyle f(t)={\frac {6{\sqrt {3}}}{\pi \left(3+t^{2}\right)^{2}}}.}










ν
=
∞


{\displaystyle \nu =\infty }




density function:








f
(
t
)
=


1

2
π




e

−



t

2


2




.


{\displaystyle f(t)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {t^{2}}{2}}}.}






see normal distribution




^ john kruschke (2014), doing bayesian data analysis, academic press; 2 edition. isbn 0124058884
^ cite error: named reference jkb invoked never defined (see page).






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