====== 카이분포 (Chi Distribution) ======
===== 정의 =====
===== 표기 =====
===== 받침 =====
 $ x \in [ \ 0 \ , \ \infty \ ) $
===== 확률밀도함수 =====
 $ f(x) = \frac{2^{1 - \nu/2} \cdot x^{\nu - 1} \cdot e^{-x^{2} / 2}}{\Gamma \left( \frac{1}{2} \nu \right) } $

<plot>
 set title "Chi Distribution PDF"
 set size 1
 set xrange [0:10]
 set yrange [0:1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 plot (2**(0.5))*exp(-(x**2)/2)/gamma(0.5) title "1", \
  x*exp(-(x**2)/2)/gamma(1) title "2", \
  2**(-1.5)*x**(4)*exp(-(x**2)/2)/gamma(2.5) title "5", \
  2**(-4)*x**(9)*exp(-(x**2)/2)/gamma(5) title "10", \
  2**(-24)*x**(49)*exp(-(x**2)/2)/gamma(25) title "50"
</plot>
===== 누적분포함수 =====
 $ F(x) = P \left( \ \frac{1}{2} \nu \ , \ \frac{1}{2} x^{2} \ \right) $

<plot>
 set title "Chi Distribution CDF"
 set size 1
 set xrange [0:10]
 set yrange [0:1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"

 cchi(x,df1)=igamma(0.5*df1,0.5*x*x)

 plot cchi(x,1) title "df = 1", \
  cchi(x,2) title "df = 2", \
  cchi(x,5) title "df = 5", \
  cchi(x,10) title "df = 10", \
  cchi(x,50) title "df = 50"
</plot>

===== 기대값 =====
 $ E(X) = \frac{\sqrt{2} \ \Gamma \left( \frac{1}{2} (\nu + 1) \right) }{\Gamma \left( \frac{1}{2} \nu \right) } $
===== 분산 =====
 $ Var(X) = \frac{2 \left[ \Gamma \left( \frac{1}{2} \nu \right) \cdot \Gamma \left( 1 + \frac{1}{2} \nu \right) - \Gamma^{2} \left( \frac{1}{2} (\nu + 1) \right) \right]}{\Gamma^{2} \left( \frac{1}{2} \nu \right)} $
===== 왜도 =====
===== 첨도 =====

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  * [[분포]]