====== t분포 (t Distribution) ======
===== 정의 =====
 두 [[확률변수]] $Z$와 $W$가 서로 [[독립]]이고 $Z$는 [[표준정규분포]]를 $W$는 [[자유도]]가 $\nu$인 [[카이스퀘어분포]]를 따를 경우, [[확률변수]]

  * $$X = \frac{Z}{\sqrt{W / \nu}}$$

 는 [[자유도]]가 $\nu$인 [[t분포]]를 따른다.
===== 표기 =====
 $$ X \sim t(\nu) $$
----
===== 받침 =====
 $$ x \in ( \ - \infty \ , \ \infty \ ) $$
===== 확률밀도함수 =====
 $$ f(x) = \frac{\left( \frac{\nu}{\nu + x^{2}} \right)^{(1 + \nu)/2}}{\sqrt{\nu} \cdot B \left( \frac{1}{2} \nu , \frac{1}{2} \right) } $$

  * 단, $B(\alpha,\beta)$는 [[베타함수]]이다.

<plot>
 set title "t Distribution PDF"
 set size 1.0
 set xrange [-5:5]
 set yrange [0:0.5]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 f(x,v) = ((v/(v+x**2))**((1+v)/2))/(sqrt(v)*((gamma(v/2)*gamma(0.5))/(gamma(v/2+0.5))))

 plot f(x,2) title "t(2)", \
  f(x,5) title "t(5)", \
  f(x,10) title "t(10)"
</plot>
===== 누적분포함수 =====
 $$ F(x) = \frac{1}{2} + \frac{x \Gamma \left( \frac{1}{2} (\nu + 1) \right) \ _{2}F_{1} \left( \frac{1}{2}, \frac{1}{2} (\nu + 1) ; \frac{3}{2} ; -\frac{x^{2}}{\nu} \right)}{\sqrt{\pi \nu} \cdot \Gamma \left( \frac{1}{2} \nu \right) } $$

<plot>
 set title "t Distribution CDF"
 set size 1.0
 set xrange [-5:5]
 set yrange [0:1.1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"
 set key left

 ct(x,df1)=(x<0.0)?0.5*ibeta(0.5*df1,0.5,df1/(df1+x*x)):1.0-0.5*ibeta(0.5*df1,0.5,df1/(df1+x*x))

 plot ct(x,2) title "t(2)", \
  ct(x,5) title "t(5)", \
  ct(x,10) title "t(10)"
</plot>
===== 기대값 =====
 $$ E(X) = 0 $$
===== 분산 =====
 $$ Var(X) = \frac{\nu}{\nu - 2} $$
===== 왜도 =====
 $$ \gamma_{1} = 0 $$
===== 첨도 =====
 $$ \gamma_{2} = \frac{6}{\nu - 4} $$
===== 특징 =====
===== 타 분포와의 관계 =====
  * [[정규분포와 t분포 관계]]
  * [[t분포와 F분포 관계]]

----
  * [[분포]]
  * [[t분포표]]