레일리분포 (Rayleigh Distribution)

$$ X \sim Rayleigh(\sigma^2) $$

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

$$ f(x) = \frac{x \cdot e^{-x^{2}/(2 \sigma^{2})}}{\sigma^{2}} $$

$$ F(x) = 1 - e^{-x^{2}/(2 \sigma^{2})} $$

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

$$ Mdn = \sigma \sqrt{\ln(4)} $$

$$ Mo = \sigma $$

$$ Var(X) = \frac{4 - \pi}{2} \sigma^{2} $$

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

$$ \gamma_{2} = - \frac{6 \pi^{2} -24 \pi +16}{(\pi - 4)^{2}} $$

$$ \mu'_{0} = 1 $$

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

$$ \mu'_{2} = 2 \sigma^{2} $$

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

$$ \mu'_{4} = 8 \sigma^{4} $$

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

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

$$ \mu_{3} = \sqrt{\frac{\pi}{2}} (\pi - 3) \sigma^{3} $$

$$ \mu_{4} = \frac{32 - 3 \pi^{2}}{4} \sigma^{4} $$

• 분포