1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?