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레일리분포 [2012/03/21 20:30] moonrepeat 새로 만듦 |
레일리분포 [2021/03/10 21:42] |
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| - | ====== 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 ===== | ||
| - | $$ \mu_{2} = \frac{4 - \pi}{2} \sigma^{2} $$ | ||
| - | |||
| - | $$ \mu_{3} = \sqrt{\frac{\pi}{2}} (\pi - 3) \sigma^{3} $$ | ||
| - | |||
| - | $$ \mu_{4} = \frac{32 - 3 \pi^{2}}{4} \sigma^{4} $$ | ||