====== 절반 정규분포 (Half-Normal Distribution) ====
===== 정의 =====
===== 표기 =====
===== 받침 =====
 $$ x \in [ \ 0 \ , \ \infty \ ) $$
===== 확률밀도함수 =====
 $$ f(x) = \frac{2 \theta}{\pi} \exp \left[ \frac{-x^{2} \theta^{2}}{\pi} \right] $$

<plot>
 set title "Half-Normal Distribution PDF"
 set size 1.0
 set xrange [0:5]
 set yrange [0:1.5]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 f(x,y) = (2*y/pi)*exp(-(x**2)*(y**2)/pi)

 plot f(x,0.5) title "(0.5)", \
  f(x,1) title "(1)", \
  f(x,2) title "(2)"
</plot>
===== 누적분포함수 =====
 $$ F(x) = \mathrm{erf} \left( \frac{\theta x}{\sqrt{\pi}} \right) $$

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

 f(x,y) = erf((y*x)/sqrt(pi))

 plot f(x,0.5) title "(0.5)", \
  f(x,1) title "(1)", \
  f(x,2) title "(2)"
</plot>
===== 기대값 =====
 $$ E(X) = \frac{1}{\theta} $$
===== 분산 =====
 $$ Var(X) = \frac{\pi - 2}{2 \theta^{2}} $$
===== 왜도 =====
 $$ \gamma_{1} = \frac{\sqrt{2} (4 - \pi)}{(\pi - 2)^{3/2}} $$
===== 첨도 =====
 $$ \gamma_{2} = \frac{8(\pi - 3)}{(\pi - 2)^{2}} $$
===== 원적률 =====
 $$ \mu'_{1} = \frac{1}{\theta} $$


 $$ \mu'_{2} = \frac{\pi}{2 \theta^{2}} $$


 $$ \mu'_{3} = \frac{\pi}{\theta^{3}} $$


 $$ \mu'_{4} = \frac{3 \pi^{2}}{4 \theta^{4}} $$


 $$ \mu'_{k} = \pi^{(k-1)/2} \theta^{-k} \Gamma \left( \frac{1}{2} (k+1) \right) $$
===== 중심적률 =====
 $$ \mu_{2} = \frac{\pi - 2}{2 \theta^{2}} $$

 $$ \mu_{3} = \frac{4 - \pi}{2 \theta^{3}} $$

 $$ \mu_{4} = \frac{3 \pi^{2} - 4 \pi -12}{4 \theta^{3}} $$

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  * [[정규분포]]