# What is $Z = \mathbb{E}[X|\mathscr{F}]$?

1. Aug 10, 2012

### operationsres

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

Can someone please explain what $Z$ 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 $\omega \in A$ (where $\Omega$ was the x-axis and $\mathbb{R}$ 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 $E[.|\mathscr{F}]$ can give rise to a function $Z(\omega)$ that has a non-constant range over $\omega \in A$ (i.e. $Z(\omega_1) \textrm{ not necessarily equal to } Z[\omega_2]$)?

Is Z simply ANY arbitrary random variable satisfying (i) and (ii) (i.e. can the image of $Z: \Omega \rightarrow \mathbb{R}$ be arbitrary as long as (i) and (ii) hold?)?

Last edited: Aug 10, 2012