====== 기하분포 (Geometric Distribution) ======
===== 표기 =====
  * $$ X \sim Geo(p)$$
    * $$ p \in [ \ 0 \ , \ 1 \ ] $$
===== 받침 =====
 $$ x \in \{ \ 0 \ , \ 1 \ , \ 2 \ , \ \cdots \ \} $$
===== 확률밀도함수 =====
 $$ p(x) = p \ (1-p)^{x} = p \ q^{x} $$

<plot>
 set title "Geometric Distribution PMF"
 set size 1
 set yrange [0:0.9]
 set xrange [-0.5:15.5]
 set xlabel "x"
 set ylabel "p(x)"
 set format y "%.2f"

 f(x,p) = p*((1-p)**(int(x)))

 plot f(x+0.5,0.2) title "Geo(0.2)" with steps, \
  f(x+0.5,0.5) title "Geo(0.5)" with steps, \
  f(x+0.5,0.8) title "Geo(0.8)" with steps
</plot>
===== 누적분포함수 =====
 $$ F(x) = 1 - (1-p)^{x+1} = 1 - q^{x+1} $$

<plot>
 set title "Geometric Distribution CDF"
 set size 1
 set yrange [0:1.1]
 set xrange [-0.5:15.5]
 set xlabel "x"
 set ylabel "F(x)
 set format y "%.2f"

 f(x,p) = 1-(1-p)**((int(x))+1)

 plot f(x+0.5,0.2) title "Geo(0.2)" with steps, \
  f(x+0.5,0.5) title "Geo(0.5)" with steps, \
  f(x+0.5,0.8) title "Geo(0.8)" with steps
</plot>
===== 기대값 =====
 $$E(X)=\frac{1-p}{p}$$
===== 분산 =====
 $$Var(X)=\frac{1-p}{p^{2}}$$
===== 왜도 =====
 $$ \gamma_{1} = \frac{2 - p}{\sqrt{1-p}} = \frac{2-p}{\sqrt{q}} $$
===== 첨도 =====
 $$ \gamma_{2} = \frac{p^{2} - 6p + 6}{1-p} = \frac{p^{2} - 6p + 6}{q} $$
===== 특성함수 =====
 $$ \phi \ (t) = \frac{p}{1 - (1 - p) \cdot e^{ \ i t}} = \frac{p}{1 - q \cdot e^{ \ i t}} $$
===== 적률생성함수 =====
 $$M(t)=\frac{p}{1-(1-p) \cdot e^{t}} = \frac{p}{1-q \cdot e^{t}}$$
===== 원적률 =====
 $$ \mu'_{1} = \frac{1-p}{p} $$

 $$ \mu'_{2} = \frac{(2-p)(1-p)}{p^{2}} $$

 $$ \mu'_{3} = \frac{(1-p) \left[ 6+(p-6)p \right] }{p^{3}} $$

 $$ \mu'_{4} = \frac{(2-p)(1-p) \left[ 12+(p-12)p \right] }{p^{4}} $$

 $$ \mu'_{k} = p \ \mathrm{Li}_{-k} (1-p) $$

  * 단, $\mathrm{Li}_{n} (z)$는 ??함수(Polylogarithm)이다.
===== 중심적률 =====
 $$ \mu_{2} = \frac{1-p}{p^{2}} $$

 $$ \mu_{3} = \frac{(p-1)(p-2)}{p^{3}} $$

 $$ \mu_{4} = \frac{(p-1)(-p^{2} +9p -9}{p^{4}} $$

 $$ \mu_{k} = p \ \Phi \left( \ 1-p \ , \ -k \ , \ \frac{p-1}{p} \ \right) $$

  * 단, $\Phi ( \ z \ , \ s \ , \ a \ )$ 는 ??함수(Lerch Transcendent)이다.
===== 특성 =====
  * [[무기억성]]을 가진다.