====== 맥스웰 분포 (Maxwell Distribution) ======
===== 표기 =====
===== 받침 =====
 $$ x \in [ \ 0 \ , \ \infty \ ) $$
===== 확률밀도함수 =====
 $$ f(x) = \sqrt{\frac{2}{\pi}} \frac{x^{2} e^{-x^{2}/(2 \alpha^{2})}}{\alpha^{3}} $$

<plot>
 set title "Maxwell Distribution PDF"
 set size 1
 set xrange [0:6]
 set yrange [0:1.4]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 fourinvsqrtpi = 2.25675833419103
 maxwell(x,a)=fourinvsqrtpi*a**3*x*x*exp(-a*a*x*x)

 plot maxwell(x,1.5) title "1.5", \
  maxwell(x,1.0) title "1.0", \
  maxwell(x,0.5) title "0.5"
</plot>
===== 누적분포함수 =====
 $$ F(x) = erf \left( \frac{x}{\sqrt{2} \ \alpha} \right) - \frac{x^{2} e^{-x^{2}/(2 \alpha^{2})}}{\alpha} \sqrt{\frac{2}{\pi}} $$

 단, $erf(x)$는 [[오차함수]]이다.

<plot>
 set title "Maxwell Distribution CDF"
 set size 1
 set xrange [0:6]
 set yrange [0:1.1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"
 set key 5,0.2

 cmaxwell(x,a)=igamma(1.5,a*a*x*x)

 plot cmaxwell(x,1.5) title "1.5", \
  cmaxwell(x,1.0) title "1.0", \
  cmaxwell(x,0.5) title "0.5"
</plot>
===== 기대값 =====
 $$ E(X) = 2 \alpha \sqrt{\frac{2}{\pi}} $$
===== 분산 =====
 $$ Var(X) = \frac{\alpha^{2} (3 \pi -8)}{\pi} $$
===== 왜도 =====
 $$ \gamma_{1} = \frac{2 \sqrt{2} \ (5 \pi - 16)}{(3 \pi - 8)^{3/2}} $$
===== 첨도 =====
 $$ \gamma_{2} = - \frac{4(3 \pi^{2} - 40 \pi +96)}{(3 \pi - 8)^{3}} $$
===== 특성함수 =====
 $$ \phi \ (t) = i \left\{ \alpha t \sqrt{\frac{2}{\pi}} - e^{- \alpha^{2} t^{2}/2} \ (\alpha^{2} t^{2} - 1) \left[ sgn (t) \ erfi \left( \frac{\alpha \ | \ t \ |}{\sqrt{2}} \right) - i \right] \right\} $$
===== 원적률 =====
 $$ \mu'_{1} = 2 \alpha \sqrt{\frac{2}{\pi}} $$

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

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

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


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

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  * [[분포]]