맥스웰 분포 (Maxwell Distribution)

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

$$ f(x) = \sqrt{\frac{2}{\pi}} \frac{x^{2} e^{-x^{2}/(2 \alpha^{2})}}{\alpha^{3}} $$

$$ F(x) = erf \left( \frac{x}{\sqrt{2} \ \alpha} \right) - \frac{x^{2} e^{-x^{2}/(2 \alpha^{2})}}{\alpha} \sqrt{\frac{2}{\pi}} $$

단, $erf(x)$는 오차함수이다.

$$ E(X) = 2 \alpha \sqrt{\frac{2}{\pi}} $$

$$ Var(X) = \frac{\alpha^{2} (3 \pi -8)}{\pi} $$

$$ \gamma_{1} = \frac{2 \sqrt{2} \ (5 \pi - 16)}{(3 \pi - 8)^{3/2}} $$

$$ \gamma_{2} = - \frac{4(3 \pi^{2} - 40 \pi +96)}{(3 \pi - 8)^{3}} $$

$$ \phi \ (t) = i \left\{ \alpha t \sqrt{\frac{2}{\pi}} - e^{- \alpha^{2} t^{2}/2} \ (\alpha^{2} t^{2} - 1) \left[ sgn (t) \ erfi \left( \frac{\alpha \ | \ t \ |}{\sqrt{2}} \right) - i \right] \right\} $$

$$ \mu'_{1} = 2 \alpha \sqrt{\frac{2}{\pi}} $$

$$ \mu'_{2} = 3 \alpha^{2} $$

$$ \mu'_{3} = 8 \alpha^{3} \sqrt{\frac{2}{\pi}} $$

$$ \mu'_{4} = 15 \alpha^{4} $$

$$ \mu'_{k} = \frac{2^{1+k/2} \ \alpha^{k} \Gamma \left( \frac{1}{2} (3+k) \right)}{\sqrt{\pi}} $$

• 분포