절반 정규분포 (Half-Normal Distribution)

$$ x \in [ \ 0 \ , \ \infty \ ) $$

$$ f(x) = \frac{2 \theta}{\pi} \exp \left[ \frac{-x^{2} \theta^{2}}{\pi} \right] $$

$$ F(x) = \mathrm{erf} \left( \frac{\theta x}{\sqrt{\pi}} \right) $$

$$ E(X) = \frac{1}{\theta} $$

$$ Var(X) = \frac{\pi - 2}{2 \theta^{2}} $$

$$ \gamma_{1} = \frac{\sqrt{2} (4 - \pi)}{(\pi - 2)^{3/2}} $$

$$ \gamma_{2} = \frac{8(\pi - 3)}{(\pi - 2)^{2}} $$

$$ \mu'_{1} = \frac{1}{\theta} $$

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

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

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

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

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

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

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

• 정규분포