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

[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?)?

 

[lippyka]

A conditional expectation is a random variable that has the same expected value as the original random variable over all events in the specified sigma-field. In this case, Z is any random variable that satisfies both (i) and (ii). In other words, the image of Z (its range) can be arbitrary as long as it satisfies (i) and (ii). This means that Z does not necessarily have to have a constant range over all \omega \in A, as long as its expected value over all events in \mathscr{F} is the same as that of X.