====== 대수정규분포 (Lognormal Distribution) ======
===== 표기 =====
 [[확률변수]] $Y = \ln X$가 [[정규분포]] $N(\mu, \sigma^{2})$을 따를 때 $X$는 [[대수정규분포]]를 따르고 아래와 같이 표기 한다.

 $$ X \sim LN(\mu , \sigma^{2})$$
===== 받침 =====
 $$ x \in ( \ 0 \ , \ \infty \ ) $$
===== 확률밀도함수 =====
 $$ f(x) = \frac{1}{\sqrt{2 \pi} \ \sigma \cdot x} \exp \left[ - \frac{(\ln x - \mu)^{2}}{2 \sigma^{2}} \right] $$

<plot>
 set title "Lognormal Distribution PDF"
 set size 1.0
 set xrange [0:8]
 set yrange [0:1.2]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 f(x,m,s) = (1/(sqrt(2*pi*s)*x))*exp(-((log(x)-m)**2)/(2*s))

 plot f(x,0,1) title "LN(0,1)", \
  f(x,0,4) title "LN(0,4)", \
  f(x,0,0.5) title "LN(0,0.5)", \
  f(x,1,1) title "LN(1,1)"
</plot>
===== 누적분포함수 =====
 $$ F(x) = \frac{1}{2} \left[ 1 + \mathrm{erf} \left( \frac{\ln x - \mu}{\sigma \sqrt{2}} \right) \right] $$

<plot>
 set title "Lognormal Distribution CDF"
 set size 1.0
 set xrange [0:8]
 set yrange [0:1.1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"

 f(x,m,s) = 0.5*(1+erf((log(x)-m)/sqrt(2*s)))

 plot f(x,0,1) title "LN(0,1)", \
  f(x,0,4) title "LN(0,4)", \
  f(x,0,0.5) title "LN(0,0.5)", \
  f(x,1,1) title "LN(1,1)"
</plot>
===== 기대값 =====
 $$ E(x) = e^{\mu + \sigma^{2}/2} $$
===== 분산 =====
 $$ Var(x) = e^{2 \mu + \sigma^{2}} \ (e^{\sigma^{2}} - 1}) $$
===== 왜도 =====
 $$ \gamma_{1} = \sqrt{e^{\sigma^{2}} - 1} \left( 2 + e^{\sigma^{2}} \right) $$
===== 첨도 =====
 $$ \gamma_{2} = e^{4 \sigma^{2}} + 2 e^{3 \sigma^{2}} + 3 e^{2 \sigma^{2}} - 6 $$
===== 원적률 =====
 $$ \mu'_{1} = e^{\mu + \sigma^{2}/2} $$

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

 $$ \mu'_{3} = e^{3 \mu + 9 \sigma^{2}/2} $$

 $$ \mu'_{4} = e^{4 \mu + 8 \sigma^{2}} $$
===== 중심적률 =====
 $$ \mu_{2} = e^{2 \mu + \sigma^{2}} \left( e^{\sigma^{2}} - 1 \right) $$

 $$ \mu_{3} = e^{3 \mu + 3 \sigma^{2}/2} \left( e^{\sigma^{2}} - 1 \right) \left( e^{\sigma^{2}} + 2 \right) $$

 $$ \mu_{4} = e^{4 \mu + 2 \sigma^{2}} \left( e^{\sigma^{2}} - 1 \right) \left( e^{4 \sigma^{2}} + 2 e^{3 \sigma^{2}} + 3 e^{2 \sigma^{2}} - 3 \right) $$

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  * [[분포]]
  * [[정규분포]]