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Homework Help: What is [itex]Z = \mathbb{E}[X|\mathscr{F}][/itex]?

  1. Aug 10, 2012 #1
    A random variable Z is called the conditional expectation of X given the sigma-field [itex]\mathscr{F}[/itex] (we write [itex]Z = E[X|\mathscr{F}][/itex]) when (i) [itex]\sigma(Z) \subset \mathscr{F}[/itex] and (ii) [itex]E[X*I_A] = E[Z*I_A] \forall A \in \mathscr{F}[/itex].

    Can someone please explain what [itex]Z[/itex] is? (Yes I've googled and looked at 2 textbooks - things I find are a little too advanced for me).

    My lecturer drew a diagram and stated that the integral over all [itex]\omega \in A[/itex] (where [itex]\Omega[/itex] was the x-axis and [itex]\mathbb{R}[/itex] was the y-axis) simply had to be equal for Z and X (i.e. area under both random variables over the event had to be equal) for (ii) to be satisfied. This makes sense given what (ii) is, but I don't see how an expectation [itex]E[.|\mathscr{F}][/itex] can give rise to a function [itex]Z(\omega)[/itex] that has a non-constant range over [itex]\omega \in A[/itex] (i.e. [itex]Z(\omega_1) \textrm{ not necessarily equal to } Z[\omega_2][/itex])?

    Is Z simply ANY arbitrary random variable satisfying (i) and (ii) (i.e. can the image of [itex]Z: \Omega \rightarrow \mathbb{R}[/itex] be arbitrary as long as (i) and (ii) hold?)?
    Last edited: Aug 10, 2012
  2. jcsd
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