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다음 판 		이전 판
기하_분포 [2011/11/28 14:57]
moonrepeat 새로 만듦 		기하_분포 [2021/03/10 21:42] (현재)
줄 1: 줄 1:
-== [기하분포] (Geometric Distribution) == +====== 기하분포 (Geometric Distribution) ====== 
----- +===== 표기 ​===== 
-=== 표기 === +  ​* ​$$ X \sim Geo(p)$$ 
- $$ X \sim Geo(p)$$ +    ​* ​$$ p \in [ \ 0 \ , \ 1 \ ] $$ 
- +===== 받침 ​=====
- +
-   $$ p \in [ \ 0 \ , \ 1 \ ] $$ +
----- +
-=== [받침] ===+
  $$ x \in \{ \ 0 \ , \ 1 \ , \ 2 \ , \ \cdots \ \} $$  $$ x \in \{ \ 0 \ , \ 1 \ , \ 2 \ , \ \cdots \ \} $$
----- +===== 확률밀도함수 ​=====
-=== [확률밀도함수] ===+
  $$ p(x) = p \ (1-p)^{x} = p \ q^{x} $$  $$ p(x) = p \ (1-p)^{x} = p \ q^{x} $$
  
- +<​plot>​
- ​{{{#​!gnuplot+
  set title "​Geometric Distribution PMF"  set title "​Geometric Distribution PMF"
- set size 0.7+ set size 1
  set yrange [0:0.9]  set yrange [0:0.9]
  set xrange [-0.5:15.5]  set xrange [-0.5:15.5]
줄 28: 줄 22:
   f(x+0.5,​0.5) title "​Geo(0.5)"​ 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   f(x+0.5,​0.8) title "​Geo(0.8)"​ with steps
-}}} +</​plot>​ 
----- +===== 누적분포함수 ​=====
-=== [누적분포함수] ===+
  $$ F(x) = 1 - (1-p)^{x+1} = 1 - q^{x+1} $$  $$ F(x) = 1 - (1-p)^{x+1} = 1 - q^{x+1} $$
  
- +<​plot>​
- ​{{{#​!gnuplot+
  set title "​Geometric Distribution CDF"  set title "​Geometric Distribution CDF"
- set size 0.7+ set size 1
  set yrange [0:1.1]  set yrange [0:1.1]
  set xrange [-0.5:15.5]  set xrange [-0.5:15.5]
줄 42: 줄 34:
  set ylabel "F(x)  set ylabel "F(x)
  set format y "​%.2f"​  set format y "​%.2f"​
- set key 13.5,0.2 
  
  ​f(x,​p) = 1-(1-p)**((int(x))+1)  ​f(x,​p) = 1-(1-p)**((int(x))+1)
줄 49: 줄 40:
   f(x+0.5,​0.5) title "​Geo(0.5)"​ 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   f(x+0.5,​0.8) title "​Geo(0.8)"​ with steps
-}}} +</​plot>​ 
----- +===== 기대값 ​=====
-=== [기대값] ===+
  ​$$E(X)=\frac{1-p}{p}$$  ​$$E(X)=\frac{1-p}{p}$$
----- +===== 분산 ​=====
-=== [분산] ===+
  ​$$Var(X)=\frac{1-p}{p^{2}}$$  ​$$Var(X)=\frac{1-p}{p^{2}}$$
----- +===== 왜도 ​=====
-=== [왜도] ===+
  $$ \gamma_{1} = \frac{2 - p}{\sqrt{1-p}} = \frac{2-p}{\sqrt{q}} $$  $$ \gamma_{1} = \frac{2 - p}{\sqrt{1-p}} = \frac{2-p}{\sqrt{q}} $$
----- +===== 첨도 ​=====
-=== [첨도] ===+
  $$ \gamma_{2} = \frac{p^{2} - 6p + 6}{1-p} = \frac{p^{2} - 6p + 6}{q} $$  $$ \gamma_{2} = \frac{p^{2} - 6p + 6}{1-p} = \frac{p^{2} - 6p + 6}{q} $$
----- +===== 특성함수 ​=====
-=== [특성함수] ===+
  $$ \phi \ (t) = \frac{p}{1 - (1 - p) \cdot e^{ \ i t}} = \frac{p}{1 - q \cdot e^{ \ i t}} $$  $$ \phi \ (t) = \frac{p}{1 - (1 - p) \cdot e^{ \ i t}} = \frac{p}{1 - q \cdot e^{ \ i t}} $$
----- +===== 적률생성함수 ​=====
-=== [적률생성함수] ===+
  ​$$M(t)=\frac{p}{1-(1-p) \cdot e^{t}} = \frac{p}{1-q \cdot e^{t}}$$  ​$$M(t)=\frac{p}{1-(1-p) \cdot e^{t}} = \frac{p}{1-q \cdot e^{t}}$$
----- +===== 원적률 ​=====
-=== [원적률] ===+
  $$ \mu'​_{1} = \frac{1-p}{p} $$  $$ \mu'​_{1} = \frac{1-p}{p} $$
- 
  
  $$ \mu'​_{2} = \frac{(2-p)(1-p)}{p^{2}} $$  $$ \mu'​_{2} = \frac{(2-p)(1-p)}{p^{2}} $$
- 
  
  $$ \mu'​_{3} = \frac{(1-p) \left[ 6+(p-6)p \right] }{p^{3}} $$  $$ \mu'​_{3} = \frac{(1-p) \left[ 6+(p-6)p \right] }{p^{3}} $$
- 
  
  $$ \mu'​_{4} = \frac{(2-p)(1-p) \left[ 12+(p-12)p \right] }{p^{4}} $$  $$ \mu'​_{4} = \frac{(2-p)(1-p) \left[ 12+(p-12)p \right] }{p^{4}} $$
  
 + $$ \mu'​_{k} = p \ \mathrm{Li}_{-k} (1-p) $$
  
- $$ \mu'​_{k} = p \ \operatorname{Li}_{ \ -k} (1-p) $$ +  * 단, $\mathrm{Li}_{n} (z)$는 ??​함수(Polylogarithm)이다. 
- +===== 중심적률 ​=====
- +
- +
-  ​단,&​nbsp&​nbsp $$\operatorname{Li}_{n} (z)$$ 는 ??​함수(Polylogarithm)이다. +
----- +
-=== [중심적률] ===+
  $$ \mu_{2} = \frac{1-p}{p^{2}} $$  $$ \mu_{2} = \frac{1-p}{p^{2}} $$
- 
  
  $$ \mu_{3} = \frac{(p-1)(p-2)}{p^{3}} $$  $$ \mu_{3} = \frac{(p-1)(p-2)}{p^{3}} $$
- 
  
  $$ \mu_{4} = \frac{(p-1)(-p^{2} +9p -9}{p^{4}} $$  $$ \mu_{4} = \frac{(p-1)(-p^{2} +9p -9}{p^{4}} $$
- 
  
  $$ \mu_{k} = p \ \Phi \left( \ 1-p \ , \ -k \ , \ \frac{p-1}{p} \ \right) $$  $$ \mu_{k} = p \ \Phi \left( \ 1-p \ , \ -k \ , \ \frac{p-1}{p} \ \right) $$
  
- +  * 단, $\Phi ( \ z \ , \ s \ , \ a \ )$ 는 ??​함수(Lerch Transcendent)이다. 
- +===== 특성 ​===== 
-  ​단,&​nbsp&​nbsp $$\Phi ( \ z \ , \ s \ , \ a \ )$$ 는 ??​함수(Lerch Transcendent)이다. +  * [[무기억성]]을 가진다.
----- +
-=== 특성 === +
- i. [무기억성]을 가진다.+