Mathematics Asked by Jiexiong687691 on December 6, 2020

I want to show that $mathbb {E}(mathbb {E}(Xmid mathcal {F}))=mathbb {E}X$. My thought is that

begin{align*}mathbb {E}(mathbb {E}(Xmid mathcal {F}))=sum_{Omega_{i}} frac {mathbb {E}(X;Omega_{i})}{mathbb {P}(Omega_{i})}mathbb {P}(Omega_{i})=mathbb{E}X,

end{align*}

where $Omega_{i}’s$ are either finite or infinitely countable partition of $Omega$ into disjoint sets, and each one has positive probability. I wonder if it’s always true that there exists such $Omega_{i}’s$ such that $mathcal {F}=sigma(Omega_{1},Omega_{2},ldots)$ with the desired properties? Thanks.

Not every sigma field is countably generated and not many sigma fields are generated by countable partitions.

$E(E(X|mathcal F) I_A)=EXI_A$ for all $A in mathcal F$ by the definition of $E(X|mathcal F)$. Just taking $A =Omega$ gives $E(E(X|mathcal F)) =EX$

Correct answer by Kavi Rama Murthy on December 6, 2020

Get help from others!

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

Recent Answers

- Lex on Does Google Analytics track 404 page responses as valid page views?
- haakon.io on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP