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