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절반_정규분포 [2012/07/08 17:32] moonrepeat [[중심적률]] |
절반_정규분포 [2021/03/10 21:42] |
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| - | ====== 절반 정규분포 (Half-Normal Distribution) ==== | ||
| - | ===== 정의 ===== | ||
| - | ===== 표기 ===== | ||
| - | ===== 받침 ===== | ||
| - | $$ x \in [ \ 0 \ , \ \infty \ ) $$ | ||
| - | ===== 확률밀도함수 ===== | ||
| - | $$ f(x) = \frac{2 \theta}{\pi} \exp \left[ \frac{-x^{2} \theta^{2}}{\pi} \right] $$ | ||
| - | <plot> | ||
| - | set title "Half-Normal Distribution PDF" | ||
| - | set size 1.0 | ||
| - | set xrange [0:5] | ||
| - | set yrange [0:1.5] | ||
| - | set format x "%.1f" | ||
| - | set format y "%.2f" | ||
| - | set xlabel "x" | ||
| - | set ylabel "f(x)" | ||
| - | |||
| - | f(x,y) = (2*y/pi)*exp(-(x**2)*(y**2)/pi) | ||
| - | |||
| - | plot f(x,0.5) title "(0.5)", \ | ||
| - | f(x,1) title "(1)", \ | ||
| - | f(x,2) title "(2)" | ||
| - | </plot> | ||
| - | ===== 누적분포함수 ===== | ||
| - | $$ F(x) = \mathrm{erf} \left( \frac{\theta x}{\sqrt{\pi}} \right) $$ | ||
| - | |||
| - | <plot> | ||
| - | set title "Half-Normal Distribution CDF" | ||
| - | set size 1.0 | ||
| - | set xrange [0:5] | ||
| - | set yrange [0:1.1] | ||
| - | set key 4.2,0.2 | ||
| - | set format x "%.1f" | ||
| - | set format y "%.2f" | ||
| - | set xlabel "x" | ||
| - | set ylabel "F(x)" | ||
| - | |||
| - | f(x,y) = erf((y*x)/sqrt(pi)) | ||
| - | |||
| - | plot f(x,0.5) title "(0.5)", \ | ||
| - | f(x,1) title "(1)", \ | ||
| - | f(x,2) title "(2)" | ||
| - | </plot> | ||
| - | ===== 기대값 ===== | ||
| - | $$ E(X) = \frac{1}{\theta} $$ | ||
| - | ===== 분산 ===== | ||
| - | $$ Var(X) = \frac{\pi - 2}{2 \theta^{2}} $$ | ||
| - | ===== 왜도 ===== | ||
| - | $$ \gamma_{1} = \frac{\sqrt{2} (4 - \pi)}{(\pi - 2)^{3/2}} $$ | ||
| - | ===== 첨도 ===== | ||
| - | $$ \gamma_{2} = \frac{8(\pi - 3)}{(\pi - 2)^{2}} $$ | ||
| - | ===== 원적률 ===== | ||
| - | $$ \mu'_{1} = \frac{1}{\theta} $$ | ||
| - | |||
| - | |||
| - | $$ \mu'_{2} = \frac{\pi}{2 \theta^{2}} $$ | ||
| - | |||
| - | |||
| - | $$ \mu'_{3} = \frac{\pi}{\theta^{3}} $$ | ||
| - | |||
| - | |||
| - | $$ \mu'_{4} = \frac{3 \pi^{2}}{4 \theta^{4}} $$ | ||
| - | |||
| - | |||
| - | $$ \mu'_{k} = \pi^{(k-1)/2} \theta^{-k} \Gamma \left( \frac{1}{2} (k+1) \right) $$ | ||
| - | ===== 중심적률 ===== | ||
| - | $$ \mu_{2} = \frac{\pi - 2}{2 \theta^{2}} $$ | ||
| - | |||
| - | $$ \mu_{3} = \frac{4 - \pi}{2 \theta^{3}} $$ | ||
| - | |||
| - | $$ \mu_{4} = \frac{3 \pi^{2} - 4 \pi -12}{4 \theta^{3}} $$ | ||
| - | |||
| - | ---- | ||
| - | * [[정규분포]] | ||