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Nov
9
comment How to plan a ride by several buses?
@Henning: Thanks! How do you model the schedules of bus routes into the graph?
Sep
3
comment Independence between conditional expectations
Thanks. how do you determine E[X∣X+Y]?
Sep
3
comment Determine measurability of E(X|N) or even $\sigma(E(X|N))$?
Are σ(E(X|N)) and σ(σ(X)∩N) equal, or is one a subset of the other?
Sep
3
comment Determine measurability of E(X|N) or even $\sigma(E(X|N))$?
Thanks. How do you determine E(X|N)? Some general rule?
Sep
3
comment Determine measurability of E(X|N) or even $\sigma(E(X|N))$?
how do you determine σ(E(X|N))? Are σ(E(X|N)) and σ(σ(X)∩N) equal, or one is the subset of the other?
Sep
3
comment Independence between conditional expectations
Is there relation between $\sigma(E(X|N))$ and $\sigma(\sigma(X) \cap N)$? math.stackexchange.com/questions/1420189/…
Sep
3
comment Independence between conditional expectations
right. @Domink. thanks. How about the second question? I was wondering if E(X|N) must be measurable wrt both sigma(X) and N?
Sep
3
comment Are the converses of the following special cases of conditional expectation also true?
why "both X and Y are square-integrable"?
Sep
3
comment Are the converses of the following special cases of conditional expectation also true?
Thanks. what is about X that is equivalent to E(X|N)=EX a.e., if independence is too much?
Sep
2
comment Is conditional expectation E(X|N) an a.e. equivalence class wrt N or underlying sigma algebra?
Let me put it in another way: If Y1 and Y2 are both in E(Y|N), are they equal a.e. (wrt N)?
Sep
1
comment Is conditional expectation E(X|N) an a.e. equivalence class wrt N or underlying sigma algebra?
"It's conceivable that you can specify criterion to partition E(X|N) into disjoint classes" doesn't make sense. I guess you know that the a.e. is a equivalent relation on the set of random variables, so the a.e. relation partitions the set of r.v.s. into equivalent classes. it isn't what I can change. My last comment is a well defined question
Sep
1
comment Is conditional expectation E(X|N) an a.e. equivalence class wrt N or underlying sigma algebra?
In that new measure space, can E(X|N) be several a.e. equivalent classes of N measurable functions?
Sep
1
comment Is conditional expectation E(X|N) an a.e. equivalence class wrt N or underlying sigma algebra?
In my last comment, by a.e.equivalent class, I meant for the new measure space with N as sigma algebra. so measurable functions are N measurable. In that new measure space, Is E(X|N) exactly an entire a.e. equivalent class of N measurable functions? Or several such classes? Or part of such a class?
Sep
1
comment Is conditional expectation E(X|N) an a.e. equivalence class wrt N or underlying sigma algebra?
Is E(X|N) exactly an entire a.e. equivalent class of N measurable functions? Or several such classes? Or part of such a class?
Sep
1
comment Is conditional expectation E(X|N) an a.e. equivalence class wrt N or underlying sigma algebra?
Can you show "there can exist a Y′ that equals Y a.e. but Y′ is not a member of E(X|N)"? I don't think so, because their integrals on any measurable set are same
Sep
1
comment Does $X ⊥ Y \leftrightarrow X ⊥ Y | Z$ implies $(X,Y) ⊥ Z$?
When is reverse used and when is converse?
Sep
1
comment How to formulate the requirements that a counterexample must satisfy?
@Brian. In the linked post, to construct an example, we have to choose what the three random variables $X$ $Y$ and $Z$ are, though the three statements about the random variables are given and not changeable.
Sep
1
comment How to formulate the requirements that a counterexample must satisfy?
@Brian: I should have added that $p_1, p_2$ and $p_3$ are given statements, which can't be changed. For an example, math.stackexchange.com/questions/1416468/…
Sep
1
comment How to formulate the requirements that a counterexample must satisfy?
@Brian: what do you mean by " take p2 and p3 to be the same statement"? p1, p2 and p3 are supposed to be given statements which can't be changed.
Sep
1
comment How to formulate the requirements that a counterexample must satisfy?
@Brian: Thanks. Is this a counterexample: $p_2$ and $p_3$ both are true, but $p_1$ isn't? How shall we find a counterexample that satisfies the requirement that $p_2$ and $p_3$ imply each other, although I know it means either both $p_2$ and $p_3$ are true, or neither is true.