Regression models that ignore regressors

Hi,

Suppose X, Y are independent discrete random variables with jont distribution P(X, Y). My regression model is of the form

where the random variable is the noise-term. Next, assume that we ignore our knowledge about . The we have another regression model

where is the noise term. We can express as

My first question is: How can we interpret the right hand side of the last equation? Since X and Y are independent, we have for each x

So the function g at x that has no knowledge about Y can be regarded as the conditional expectation of f given X = x. Is this argumentation correct? If independence does not hold, how do you call the expression

which looks a bit like an expectation?

The last question is concerned with the error term . From the above, we have

If I assume that , then I may conclude

But under which conditions is my assumption about Z' valid?

Thanks und best wishes,

samosa