// ㄷㄱㄱ Week 8-1
joint_probability : $\text{Pr}[X,Y]$
'''joint CDF''' : $F_{X,Y}(x,y)=\text{Pr}[X\le x,Y\le y]$

'''Joint CDF'''의 성질들
 1. $0 \le F_{X,Y}(x,y) \le 1$
 1. When $x_1 \le x_2$ and $y_1 \le y_2,$
 $F_{X,Y}(x_1,y_1) \le F_{X,Y}(x_2,y_2)$
 1. $F_{X,Y}(x,\infty)=\text{Pr}[X<x,Y<\infty]=F_X(x)$ ; marginal cdf
  ''사실 $Y<\infty$ 는 당연한 거라서 $\text{Pr}[Y<\infty]=1$ 이고 위의 $\text{Pr}[X<x,Y<\infty]=\text{Pr}[X\le x]$ 이며, 즉 X에 대한 것만이 되며, 이것을 marginal cdf 라고 한다''
   marginal_cdf는 아마 다음 페이지명으로 - [[주변누적분포함수,marginal_cumulative_distribution_function,marginal_CDF]]
   (mklink: [[주변확률,marginal_probability]]
   and [[주변확률분포,marginal_probability_distribution]]
   and [[주변확률밀도함수,marginal_probability_density_function,marginal_PDF]]
   and [[누적분포함수,cumulative_distribution_function,CDF]]
   and [[결합누적분포함수,joint_cumulative_distribution_function,joint_CDF]])
 1. $F_{X,Y}(\infty,\infty)=1,$
 $F_{X,Y}(-\infty,-\infty)=0,$
 $F_{X,Y}(-\infty,y)=?$
 $F_{X,Y}(x,-\infty)=?$
 ''즉 대충… '무한으로 뻗어갈 때' X, Y가 같은 방향이면 1, 반대 방향이면 0, 그 외의 경우는 결정불가능?''
 1. $\text{Pr}[x_1 < X \le x_2, \; y_1 < Y \le y_2]$
  $= F_{X,Y}(x_2,y_2)-F_{X,Y}(x_1,y_2) -F_{X,Y}(x_2,y_1) + F_{X,Y}(x_1,y_1)$

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두 [[확률변수,random_variable]] $X,Y$ 에 대해, '''joint CDF'''는
 $F_{X,Y}(x,y)=P(X\le x,Y\le y)$
[[http://bigdata.dongguk.ac.kr/lectures/med_stat/_book/%ED%99%95%EB%A5%A0%EB%B3%80%EC%88%98%EB%B6%84%ED%8F%AC.html src. 2.4.]]

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For 2-random variable $\mathbb{X}=(X,Y),\;-\infty<x,y<\infty,$
 $F_{X,Y}(x_1,y_1)=P_{X,Y}(X\le x_1, Y\le y_1)$
는 '''joint cdf''' of $\mathbb{X}(\text{or }X,Y)$

성질
1. CDF(__누적__분포함수)이므로 비감소함수.
 For $x_1\le x_2 \text{ and } y_1\le y_2,$
 $F_{X,Y}(x_1,y_1)\le F_{X,Y}(x_2,y_2)$
2.
 $F_{X,Y}(x_1,-\infty)=F_{X,Y}(-\infty,y_1)=0$
 and
 $F_{X,Y}(\infty,\infty)=1$
3.
 $F_X(x_1)=P(X\le x_1)=F_{X,Y}(x_1,\infty)$
 $F_Y(y_1)=P(Y\le y_1)=F_{X,Y}(\infty,y_1)$
 related; marginal_CDF
4.
 $F_{X,Y}:$ continuous from north and east
5.
 $P(x_1<X\le x_2,y_1<Y\le y_2)=F_{X,Y}(x_2,y_2)-F_{X,Y}(x_1,y_2)-F_{X,Y}(x_2,y_1)+F_{X,Y}(x_1,y_1)$
 이건 2차원 그림을 생각하면 당연.

''from 경북대 http://www.kocw.net/home/search/kemView.do?kemId=1279832 4. Joint cdf''

= links ko =
https://blastic.tistory.com/207

참고(not twins):
WpKo:결합분포
WpEn:Joint_probability_distribution

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Up: [[결합,joint]] [[누적분포함수,cumulative_distribution_function,CDF]]