====== 레일리분포 (Rayleigh Distribution) ======
===== 정의 =====
===== 표기 =====
 $$ X \sim Rayleigh(\sigma^2) $$
===== 받침 =====
 $$ x \in [ \ 0 \ , \ \infty \ ) $$
===== 확률밀도함수 =====
 $$ f(x) = \frac{x \cdot e^{-x^{2}/(2 \sigma^{2})}}{\sigma^{2}} $$

<plot>
 set title "Rayleigh Distribution PDF"
 set size 1
 set xrange [0:10]
 set yrange [0:1.3]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 f(x,s) = (x*exp(-(x**2)/(2*(s**2))))/(s**2)

 plot f(x,0.5) title "(0.5)", \
  f(x,1.0) title "(1.0)", \
  f(x,2.0) title "(2.0)", \
  f(x,3.0) title "(3.0)", \
  f(x,4.0) title "(4.0)"
</plot>
===== 누적분포함수 =====
 $$ F(x) = 1 - e^{-x^{2}/(2 \sigma^{2})} $$

<plot>
 set title "Rayleigh Distribution CDF"
 set size 1
 set xrange [0:10]
 set yrange [0:1.1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"

 f(x,s) = 1-exp(-(x**2)/(2*(s**2)))

 plot f(x,0.5) title "(0.5)", \
  f(x,1.0) title "(1.0)", \
  f(x,2.0) title "(2.0)", \
  f(x,3.0) title "(3.0)", \
  f(x,4.0) title "(4.0)"
</plot>
===== 기대값 =====
 $$ E(X) = \sigma \sqrt{\frac{\pi}{2}} $$
===== 중앙값 =====
 $$ Mdn = \sigma \sqrt{\ln(4)} $$
===== 최빈값 =====
 $$ Mo = \sigma $$
===== 분산 =====
 $$ Var(X) = \frac{4 - \pi}{2} \sigma^{2} $$
===== 왜도 =====
 $$ \gamma_{1} = \frac{2(\pi - 3) \sqrt{\pi}}{(4 - \pi)^{3/2}} $$
===== 첨도 =====
 $$ \gamma_{2} = - \frac{6 \pi^{2} -24 \pi +16}{(\pi - 4)^{2}} $$
===== 원적률 =====
 $$ \mu'_{0} = 1 $$

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

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

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

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

 $$ \mu'_{k} = 2^{k/2} \cdot \sigma^{k} \cdot \Gamma \left( 1 + \frac{1}{2} k \right) $$
===== 중심적률 =====
 $$ \mu_{2} = \frac{4 - \pi}{2} \sigma^{2} $$

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

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

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  * [[분포]]