====== 라플라스분포 (Laplace Distribution) ======
===== 정의 =====
===== 표기 =====
===== 받침 =====
 $$ x \in ( \ -\infty \ , \ \infty \ ) $$
===== 확률밀도함수 =====
 $$ f(x) = \frac{1}{2b} \exp \left[ \frac{- | \ x - \mu \ | }{b} \right] $$

<plot>
 set title "Laplace Distribution PDF"
 set size 1
 set xrange [-5:5]
 set yrange [0:1.4]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 f(x,m,b) = (1/(2*b))*exp(-(abs(x-m))/b)

 plot f(x,0,0.5) title "(0,1/2)", \
  f(x,0,0.333) title "(0,1/3)", \
  f(x,1,0.5) title "(1,1/2)"
</plot>
===== 누적분포함수 =====
 $$ F(x) = \frac{1}{2} \left\{ 1 + sgn (x - \mu) \left( 1 - \exp \left[ \frac{- | \ x - \mu \ | }{b} \right] \right) \right\} $$

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

 f(x,m,b) = (0.5)*(1+(sgn(x-m))*(1-exp(-(abs(x-m))/b)))

 plot f(x,0,0.5) title "(0,1/2)", \
  f(x,0,0.333) title "(0,1/3)", \
  f(x,1,0.5) title "(1,1/2)"
</plot>
===== 기대값 =====
 $$ E(X) = \mu $$
===== 분산 =====
 $$ Var(X) = 2 b^{2} $$
===== 왜도 =====
 $$ \gamma_{1} = 0 $$
===== 첨도 =====
 $$ \gamma_{2} = 3 $$
===== 특성함수 =====
 $$ \phi \ (t) = \frac{e^{i \mu t}}{1 + b^{2} t^{2}} $$