E{XE{Y|C}}=E{YE{X|C}}

现在有没有适合大学生用的搜题软件呢?  https://www.zhihu.com/question/51935291/answer/514312093  Approach0 (https://approach0.xyz/search/) 专门用来搜索 Math StackExchange (支持数学公式)。

the if direction:

https://math.stackexchange.com/questions/2494959/x-y-almost-surely-iff-forall-a-in-mathcalg-int-axdp-int-aydp?r=SearchResults&s=4|45.8271

zhihu search:    math stack exchange

       math exchange

2,slader: 这个软件是帮助你写作业的,一般的教科书后面会有答案,但不会有详解,这个软件会帮助你更好的学习。

手机上当然首推Desmos,Wolfram Alpha还有Brilliant啦!

Math Lab。能解微分方程能画心型图神器

设x,y是概率空间(Ω,F,P)上的拟可积随机变量,证明:X=Y a.e 当且仅当 xdp = ydp 对每个A∈F成立

Let (Ω,F,P)(Ω,F,P) be a probability space with G⊂FG⊂F. Let X,YX,Y be GG-measurable, and integrable. Then, how does one prove that

X=YX=Y almost surely iff ∀A∈G∫AXdP=∫AYdP∀A∈G∫AXdP=∫AYdP?

Here's my try: ∫AXdP=∫AYdP⇔∫AX−YdP=∫A(X−Y)1[X≥Y]+(Y−X)1[X<Y]dP=0∫AXdP=∫AYdP⇔∫AX−YdP=∫A(X−Y)1[X≥Y]+(Y−X)1[X<Y]dP=0

For A=[Y≥X]A=[Y≥X], we get ∫[X≥Y]X−YdP=0∫[X≥Y]X−YdP=0 which implies, by nonnegativity of X−YX−Y on A, P(X=Y)=P(X≥Y)P(X=Y)=P(X≥Y) or P(X≥Y)=0P(X≥Y)=0

For A=[Y<X]A=[Y<X], we get ∫[X<Y]Y−XdP=0∫[X<Y]Y−XdP=0 which implies, by nonnegativity of X−YX−Y on A, P(X=Y)=P(X<Y)P(X=Y)=P(X<Y) or P(X<Y)=0P(X<Y)=0

So, we get

(P(X≥Y)=0 ∧ P(X=Y)=P(X<Y)) ∨ (P(X<Y)=0 ∧ P(X=Y)=P(X≥Y))(P(X≥Y)=0 ∧ P(X=Y)=P(X<Y)) ∨ (P(X<Y)=0 ∧ P(X=Y)=P(X≥Y))

which gives P(X=Y)=1P(X=Y)=1.

I can choose A as above since the sum of measurable functions is also measurable [X>Y]=[X−Y>0][X>Y]=[X−Y>0]

Is this a proper proof?

Who is BB, and what is meant by "Let X,YX,Y be random variables in GG?" That XX and YY are σ(G)σ(G)-measurable? – Math1000 Oct 30 '17 at 10:41

现在有没有适合大学生用的搜题软件呢?

 
05-18 05:10