====== 베타분포 (Beta Distribution) ======
===== 정의 =====
===== 표기 =====
 $$ X \sim Be(\alpha , \beta)$$

  * $$ \alpha \in ( \ 0 \ , \ \infty \ ) $$
  * $$ \beta \in ( \ 0 \ , \ \infty \ ) $$
===== 받침 =====
 $$ x \in [ \ 0 \ , \ 1 \ ] $$
===== 확률밀도함수 =====
 $$f(x)= \left[ \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} \right] x^{a-1}(1-x)^{\beta-1} $$

 (1) 그래프 : $\alpha \ \mathrm{or} \ \beta \ = \ 1$

 (2) 그래프 : $\alpha \ = \ \beta \ < \ 1$

 (3) 그래프 : $\alpha \ = \ \beta > 1$

 (4) 그래프 : $1 \ < \ \alpha \ < \ \beta \ \mathrm{or} \ 1 \ < \ \beta \ < \ \alpha$

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

 f(x,a,b) = (gamma(a+b)/(gamma(a)*gamma(b)))*(x**(a-1))*((1-x)**(b-1))

 plot f(x,1,1) title "Be(1,1)", \
  f(x,0.5,1) title "Be(0.5,1)", \
  f(x,1,0.5) title "Be(1,0.5)", \
  f(x,2,1) title "Be(2,1)", \
  f(x,1,2) title "Be(1,2)", \
  f(x,4,1) title "Be(4,1)", \
  f(x,1,4) title "Be(1,4)"
</plot>
<plot>
 set title "Beta Distribution PDF (2)"
 set size 1.0
 set xrange [0:1]
 set yrange [0:5]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 f(x,a,b) = (gamma(a+b)/(gamma(a)*gamma(b)))*(x**(a-1))*((1-x)**(b-1))

 plot f(x,0.9,0.9) title "Be(0.9,0.9)", \
  f(x,0.7,0.7) title "Be(0.7,0.7)", \
  f(x,0.5,0.5) title "Be(0.5,0.5)", \
  f(x,0.3,0.3) title "Be(0.3,0.3)", \
  f(x,0.1,0.1) title "Be(0.1,0.1)"
</plot>

<plot>
 set title "Beta Distribution PDF (3)"
 set size 1.0
 set xrange [0:1]
 set yrange [0:5]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 f(x,a,b) = (gamma(a+b)/(gamma(a)*gamma(b)))*(x**(a-1))*((1-x)**(b-1))

 plot f(x,2,2) title "Be(2,2)", \
  f(x,4,4) title "Be(4,4)", \
  f(x,6,6) title "Be(6,6)", \
  f(x,8,8) title "Be(8,8)", \
  f(x,10,10) title "Be(10,10)"
</plot>
<plot>
 set title "Beta Distribution PDF (4)"
 set size 1.0
 set xrange [0:1]
 set yrange [0:5]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 f(x,a,b) = (gamma(a+b)/(gamma(a)*gamma(b)))*(x**(a-1))*((1-x)**(b-1))

 plot f(x,2,4) title "Be(2,4)", \
  f(x,2,6) title "Be(2,6)", \
  f(x,2,8) title "Be(2,8)", \
  f(x,4,2) title "Be(4,2)", \
  f(x,6,2) title "Be(6,2)", \
  f(x,8,2) title "Be(8,2)"
</plot>
===== 누적분포함수 =====
 $$ F(x) = I( \ x \ ; \ \alpha \ , \ \beta \ ) $$

  * 단, $I( \ x \ ; \ \alpha \ , \ \beta \ )$는 [[정칙 베타함수]]이다.

 (1) 그래프 : $\alpha \ \mathrm{or} \ \beta \ = \ 1$

 (2) 그래프 : $\alpha \ = \ \beta \ < \ 1$

 (3) 그래프 : $\alpha \ = \ \beta > 1$

 (4) 그래프 : $1 \ < \ \alpha \ < \ \beta \ \mathrm{or} \ 1 \ < \ \beta \ < \ \alpha$

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

 f(x,a,b) = ibeta(a,b,x)

 plot f(x,1,1) title "Be(1,1)", \
  f(x,0.5,1) title "Be(0.5,1)", \
  f(x,1,0.5) title "Be(1,0.5)", \
  f(x,2,1) title "Be(2,1)", \
  f(x,1,2) title "Be(1,2)", \
  f(x,4,1) title "Be(4,1)", \
  f(x,1,4) title "Be(1,4)"
</plot>
<plot>
 set title "Beta Distribution CDF (2)"
 set size 1.0
 set xrange [0:1]
 set yrange [0:1.1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"

 f(x,a,b) = ibeta(a,b,x)

 plot f(x,0.9,0.9) title "Be(0.9,0.9)", \
  f(x,0.7,0.7) title "Be(0.7,0.7)", \
  f(x,0.5,0.5) title "Be(0.5,0.5)", \
  f(x,0.3,0.3) title "Be(0.3,0.3)", \
  f(x,0.1,0.1) title "Be(0.1,0.1)"
</plot>

<plot>
 set title "Beta Distribution CDF (3)"
 set size 1.0
 set xrange [0:1]
 set yrange [0:1.1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"

 f(x,a,b) = ibeta(a,b,x)

 plot f(x,2,2) title "Be(2,2)", \
  f(x,4,4) title "Be(4,4)", \
  f(x,6,6) title "Be(6,6)", \
  f(x,8,8) title "Be(8,8)", \
  f(x,10,10) title "Be(10,10)"
</plot>
<plot>
 set title "Beta Distribution CDF (4)"
 set size 1.0
 set xrange [0:1]
 set yrange [0:1.1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"

 f(x,a,b) = ibeta(a,b,x)

 plot f(x,2,4) title "Be(2,4)", \
  f(x,2,6) title "Be(2,6)", \
  f(x,2,8) title "Be(2,8)", \
  f(x,4,2) title "Be(4,2)", \
  f(x,6,2) title "Be(6,2)", \
  f(x,8,2) title "Be(8,2)"
</plot>
===== 기대값 =====
 $$E(X)=\frac{\alpha}{\alpha+\beta}$$
===== 분산 =====
 $$Var(X)=\frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}$$
===== 왜도 =====
 $$ \gamma_{1} = \frac{2(\beta - \alpha) \sqrt{1 + \alpha + \beta}}{\sqrt{\alpha \beta} (2 + \alpha + \beta)} $$
===== 첨도 =====
 $$ \gamma_{2} = \frac{6 \left[ \alpha^{3} + \alpha^{2} (1 - 2 \beta) + \beta^{2} (1 + \beta) - 2 \alpha \beta (2 + \beta) \right] }{\alpha \beta (\alpha + \beta + 2) (\alpha + \beta + 3)} $$
===== 특성함수 =====
 $$ \phi \ (t) = \ _{1}F_{1} (\alpha \ ; \ \alpha + \beta \ ; \ i \cdot t) $$
===== 원적률 =====
 $$ \mu'_{k} = \frac{\Gamma (\alpha + \beta) \cdot \Gamma (\alpha + k)}{\Gamma (\alpha + \beta + k) \cdot \Gamma (\alpha)} $$
===== 중심적률 =====
 $$ \mu_{k} = \left( - \frac{\alpha}{\alpha + \beta} \right)^{k} \ _{2}F_{1} \left( -k \ , \ \alpha \ ; \ \alpha + \beta \ ; \ \frac{\alpha + \beta}{\alpha} \right) $$

----
  * [[분포]]
  * [[연속형 분포]]