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I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, please let me know.

Let me get more specific. Let $\left(\Omega,\mathcal{A},P\right)$ be a probability space and $X$ an integrable real random variable defined on $(\Omega,\mathcal{A},P)$. Let $\mathcal{F}$ be a sub-$\sigma$-algebra of $\mathcal{A}$. Then $E[X|\mathcal{F}]$ is the a.s. unique random variable $Y$ such that $Y$ is $\mathcal{F}$-measurable and for any $A\in\mathcal{F}$, $E\left[X1_A\right]=E\left[Y1_A\right]$.

The common interpretation seems to be: "$E[X|\mathcal{F}]$ is the expectation of $X$ given the information of $\mathcal{F}$." I'm finding it hard to get any meaning from this sentence.

  1. In elementary probability theory, expectation is a real number. So the sentence above makes me think of a real number instead of a random variable. This is reinforced by $E[X|\mathcal{F}]$ sometimes being called "conditional expected value". Is there some canonical way of getting real numbers out of $E[X|\mathcal{F}]$ that can be interpreted as elementary expected values of something?

  2. In what way does $\mathcal{F}$ provide information? To know that some event occurred, is something I would call information, and I have a clear picture of conditional expectation in this case. To me $\mathcal{F}$ is not a piece of information, but rather a "complete" set of pieces of information one could possibly acquire in some way.

Maybe you say there is no real intuition behind this, $E[X|\mathcal{F}]$ is just what the definition says it is. But then, how does one see that a martingale is a model of a fair game? Surely, there must be some intuition behind that!

I hope you have got some impression of my misconceptions and can rectify them.

  • 2
    This is not the definition of conditional expectation with which I'm familiar. Do you have a reference?2011-02-24
  • 1
    @Qiaochu: I'm using Klenke's [Probability Theory](http://books.google.de/books?id=tcm3y5UJxDsC&printsec=frontcover&dq=probability+theory&hl=de&ei=RcpmTba0Ic_Ysgbxu83uDA&sa=X&oi=book_result&ct=result&resnum=4&ved=0CEcQ6AEwAw#v=onepage&q&f=false), but it's the same on Wikipedia.2011-02-24
  • 0
    You may want to read the answer to this question, http://math.stackexchange.com/questions/23093/could-someone-explain-conditional-independence/23100#23100, where user joriki explains what it means for event A to be conditionally dependent on event B.2011-02-24
  • 0
    A wonderful explanation about conditional expectation can be found here https://www.ma.utexas.edu/users/gordanz/notes/conditional_expectation.pdf2016-08-12

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