Characterization Student's t-distribution

1 characterization

1.1 distribution of test statistic

1.1.1 derivation


1.2 maximum entropy distribution

characterization
as distribution of test statistic

student s t-distribution



ν


{\displaystyle \nu }

degrees of freedom can defined distribution of random variable t with

t
=


z

v

/

ν



=
z



ν
v



,


{\displaystyle t={\frac {z}{\sqrt {v/\nu }}}=z{\sqrt {\frac {\nu }{v}}},}

where

z standard normal expected value 0 , variance 1;
v has chi-squared distribution



ν


{\displaystyle \nu }

degrees of freedom;
z , v independent.

a different distribution defined of random variable defined, given constant μ, by

(
z
+
μ
)



ν
v



.


{\displaystyle (z+\mu ){\sqrt {\frac {\nu }{v}}}.}

this random variable has noncentral t-distribution noncentrality parameter μ. distribution important in studies of power of student s t-test.

derivation

suppose x1, ..., xn independent realizations of normally-distributed, random variable x, has expected value μ , variance σ. let

x
¯



n


=


1
n


(

x

1


+
⋯
+

x

n


)


{\displaystyle {\overline {x}}_{n}={\frac {1}{n}}(x_{1}+\cdots +x_{n})}

be sample mean, and

s

n


2


=


1

n
−
1




∑

i
=
1


n




(

x

i


−



x
¯



n


)


2




{\displaystyle s_{n}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}_{n}\right)^{2}}

be unbiased estimate of variance sample. can shown random variable

v
=
(
n
−
1
)



s

n


2



σ

2






{\displaystyle v=(n-1){\frac {s_{n}^{2}}{\sigma ^{2}}}}

has chi-squared distribution



ν
=
n
−
1


{\displaystyle \nu =n-1}

degrees of freedom (by cochran s theorem). readily shown quantity

z
=

(



x
¯



n


−
μ
)




n

σ




{\displaystyle z=\left({\overline {x}}_{n}-\mu \right){\frac {\sqrt {n}}{\sigma }}}

is distributed mean 0 , variance 1, since sample mean






x
¯



n




{\displaystyle {\overline {x}}_{n}}

distributed mean μ , variance σ/n. moreover, possible show these 2 random variables (the distributed 1 z , chi-squared-distributed 1 v) independent. consequently pivotal quantity

t
≡


z

v

/

ν



=

(



x
¯



n


−
μ
)




n


s

n




,


{\displaystyle t\equiv {\frac {z}{\sqrt {v/\nu }}}=\left({\overline {x}}_{n}-\mu \right){\frac {\sqrt {n}}{s_{n}}},}

which differs z in exact standard deviation σ replaced random variable sn, has student s t-distribution defined above. notice unknown population variance σ not appear in t, since in both numerator , denominator, canceled. gosset intuitively obtained probability density function stated above,



ν


{\displaystyle \nu }

equal n − 1, , fisher proved in 1925.

the distribution of test statistic t depends on



ν


{\displaystyle \nu }

, not μ or σ; lack of dependence on μ , σ makes t-distribution important in both theory , practice.

as maximum entropy distribution

student s t-distribution maximum entropy probability distribution random variate x



e
(
ln
⁡
(
ν
+

x

2


)
)


{\displaystyle e(\ln(\nu +x^{2}))}

fixed.