Tim
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 Apr 28 comment What formal systems are various programming paradigms based on? Thanks. Are automatons (e.g. Turing machine, finite state machine) formal systems? Is procedural paradigm (or more generally, imperative paradigm, which consist of both procedural and OO paradigms) based on automatons? 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.