#noindex
Sub:
[[확률공간,probability_space]]

REL
[[확률변수,random_variable]]

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Probability law ''P''라는 것은, ?를 정의역으로 하는 함수
 $P:?\to[0,1]$

?가 무엇인지 확실히
 [[사건,event]]?

Sub:
 [[사건,event]]
 [[결과,outcome]]
 [[분할,partition]]
 [[전확률,total_probability]] (전확률정리)
 [[베이즈_정리,Bayes_s_theorem]]
 [[베이즈_확률,Bayesian_probability]] ?
{
MKL
[[베이즈_정리,Bayes_theorem]] - [[VG:베이즈_정리,Bayes_theorem]]

'''Bayesian probability'''

주관적 [[믿음,belief]]을 반영한?

WtEn:Bayesian_probability x [[Date(2024-01-12T05:55:39)]]
WpSp:Bayesian_probability ?
WpEn:Bayesian_probability ?

Ndict:"베이즈 확률"
Ggl:"베이즈 확률"
Ggl:"Bayesian probability"
}
 [[조건부확률,conditional_probability]]
{
[[결합확률,joint_probability]] 관련.
}
 [[결합확률,joint_probability]] =결합확률,joint_probability =,joint_probability . joint_probability 결합확률
{
joint probability

동시확률? 결합확률?
결합확률 via KmsE:"joint probability"

WtEn:joint_probability = https://en.wiktionary.org/wiki/joint_probability

복수의 확률 변수에 대해서 그들의 동시 발생을 확률적으로 나타낸 양.
$\mathbf{P}[A\cap B]=\mathbf{P}[B|A]\mathbf{P}[A]=\mathbf{P}[A|B]\mathbf{P}[B]$

[[조건부확률,conditional_probability]]을 써야 표현됨.

"joint probability"
Ndict:"joint probability"
Ggl:"joint probability"
Ndict:결합확률
Ggl:결합확률

Rel
[[확률분포,probability_distribution]]관련하여:
 [[결합확률분포,joint_probability_distribution]]
 [[VG:결합확률분포,joint_probability_distribution]]
[[확률밀도함수,probability_density_function,PDF]]관련하여:
 [[결합확률밀도함수,joint_probability_density_function,joint_PDF]] <- 이것들 소문자화할까? pdf 등
 [[VG:결합확률밀도함수,joint_probability_density_function,joint_PDF]]
[[확률질량함수,probability_mass_function,PMF]]관련하여:
 [[결합확률질량함수,joint_probability_mass_function,joint_PMF]]
 [[VG:결합확률질량함수,joint_probability_mass_function,joint_PMF]]

wikiadmin 
Up: [[조인트,joint]] [[결합,joint]]

[[VG:결합확률,joint_probability]]

}
 [[확률분포,probability_distribution]]
 …
 … Self:probability_ Self:probabilistic

= Probability axioms =
||$P(A)\ge 0$ ||nonnegativity ||
||$P(A\cup B)=P(A)+P(B)\textrm{ for }A\cap B=\not\bigcirc$ ||countable additivity. ([[https://mathworld.wolfram.com/CountableAdditivity.html 참고]]) ||
||$P(\Omega)=1$ ||normalization? ||

= Total probability theorem =
$A_1,\cdots,A_n$ : partition of Ω
$P(B)=P(A_1)P(B|A_1)+P(A_2)P(B|A_2)+\cdots+P(A_n)P(B|A_n)$
 $=P\left(\bigcup_{i=1}^{n}(B\cap A_i)\right)$
 $=P(B\cap A_1)+\cdots+P(B\cap A_n)$

[[VG:전확률정리,total_probability_theorem]]

= Bayes' theorem =
$A_1,\cdots,A_n$ : partition of Ω
 with $P(A_i)>0\;\;\forall i$

$P[A_i|B]$
 $=\frac{P[A_i\cap B]}{P[B]}$
 $=\frac{P[A_i]\cdot P[B|A_i]}{P[A_1]\cdot P[B|A_1]+\cdots+P[A_n]\cdot P[B|A_n]}$

[[베이즈_정리,Bayes_s_theorem]]
[[VG:베이즈_정리,Bayes_theorem]]

== ex. radar detection ==
A = {aircraft present, 비행기 나타남}
B = {alarm, 경보 울림}
P(A)=0.05
P(B|A)=0.99 // 비행가기 나타나면 경보 울릴 확률이 0.99
 이것만 보면 굉장히 정확해 보인다.
P(B|A^^C^^)=0.1 // 비행기가 없는데 경보 울릴 확률은 0.1
Q: 경보가 울렸을 때 실제로 비행기가 존재할 확률?
P(A|B)
 $=\frac{P(A\cap B)}{P(B)}$
 $=\frac{P(A)P(B|A)}{P(A)P(B|A)+P(A^c)P(B|A^c)}$
 $=\frac{0.05\times 0.99}{0.05\times 0.99+0.95\times 0.1} \simeq 0.3426$

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...(계속?) 확률 용어
[[확률실험,random_experiment]]
{
실험을 하면 결과(result)로 [[결과,outcome]]가 나온다.
가능한 모든 결과의 집합: [[표본공간,sample_space]]

}
[[표본공간,sample_space]]

[[상호배타적,mutually_exclusive]] (서로 배반)
[[전체포괄적,collectively_exhaustive]]

[[VG:순열,permutation]]
{
뽑아서 늘어놓는다. 늘어놓는 순서가 중요하다.
${}_n\mathrm{P}_{r}=\frac{n!}{(n-r)!}$
}

[[VG:조합,combination]]
{
표기: ${}_{n}\mathrm{C}_{r}= \binom{n}{r}$
뽑기만 하고 순서는 중요치 않다.
${}_n\mathrm{C}_{r}=\frac{n!}{r!(n-r)!}$
}


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// from http://contents.kocw.or.kr/KOCW/document/2014/sookmyung/yeoinkwon/4.pdf p.70
확률의 정리들 (easy)
 * $P(A^C)=1-P(A)$
 * $A\subset B \Rightarrow P(A)\le P(B)$
 * $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
 * $P(A\cup B)\le P(A)+P(B)$
 * $P(A\cap B)=P(A)P(B|A)=P(B)P(A|B)$
 * $P(B)=P(A\cap B)+P(A^C\cap B)=P(A)P(B|A)+P(A^C)P(B|A^C)$
  [[VG:전확률정리,total_probability_theorem]]에서 n=2인 경우 맞는지 chk

= 확률 + 통계 공통내용이 많은데.... =
REL '''확률''' with [[통계,statistics]] / [[통계학,statistics]]

[[VG:확률과_통계,probability_and_statistics]]

[[MathNote:확률과_통계]] = https://wiki.mathnt.net/index.php?title=확률과_통계
[[MathNote:확률과_통계_관련_주제들]] = https://wiki.mathnt.net/index.php?title=확률과_통계_관련_주제들

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Twins:

[[https://terms.naver.com/entry.naver?docId=3405184&cid=47324&categoryId=47324 수학백과: 수학적 확률]] (i.e. 확률의 고전적 정의)

[[VG:확률,probability]]

http://biohackers.net/wiki/Probability

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Up:
[[수학,math]] esp [[확률론,probability_theory]]
[[확률및랜덤프로세스,probability_and_random_process]]