meta data for this page
차이
문서의 선택한 두 판 사이의 차이를 보여줍니다.
양쪽 이전 판 이전 판 다음 판 | 이전 판 | ||
기하_분포 [2017/08/07 16:18] moonrepeat |
기하_분포 [2021/03/10 21:42] (현재) |
||
---|---|---|---|
줄 1: | 줄 1: | ||
====== 기하분포 (Geometric Distribution) ====== | ====== 기하분포 (Geometric Distribution) ====== | ||
- | ---- | ||
===== 표기 ===== | ===== 표기 ===== | ||
- | $$ X \sim Geo(p)$$ | + | * $$ X \sim Geo(p)$$ |
- | + | * $$ p \in [ \ 0 \ , \ 1 \ ] $$ | |
- | + | ||
- | $$ p \in [ \ 0 \ , \ 1 \ ] $$ | + | |
- | ---- | + | |
===== 받침 ===== | ===== 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. |
- | + | ||
- | + | ||
- | + | ||
- | 단,   $$\operatorname{Li}_{n} (z)$$ 는 ??함수(Polylogarithm)이다. | + | |
- | ---- | + | |
===== 중심적률 ===== | ===== 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. | |
- | + | ||
- | 단,   $$\Phi ( \ z \ , \ s \ , \ a \ )$$ 는 ??함수(Lerch Transcendent)이다. | + | |
- | ---- | + | |
===== 특성 ===== | ===== 특성 ===== | ||
- | i. [[무기억성]]을 가진다. | + | * [[무기억성]]을 가진다. |