Why do we prefer white noise and iid property? EDIT


*

*When we add noise to system we say it is white noise. White noise will have a constant power spectral density, flat power spectrum. But what is the advantage of this?

*Why do we need iid property?

*When we find out the probability density function or is it the distribution function that we need the data to be iid? Why?
Thank you
 A: Assuming you're asking "why might we use an independent, additive Gaussian assumption?" rather than "why maximize entropy?" more generally ...
I don't think it's a matter of preferring white noise or (even just additive noise) as such, though it depends on the sense in which you mean 'prefer'.
A vector is white noise if

its components each have a probability distribution with zero mean and finite variance, and are statistically independent

More specifically you ask about additive Gaussian white noise. I assume we can take zero mean and finite variance as a given, so that leaves additivity, independence and normality.


*

*identically distributed white noise is often a reasonable first approximation to reality

*often we can relatively easily do calculations under the assumption of additive independent Gaussian noise that are more difficult under other assumptions

*often the procedures we get under that assumption may still have good properties under a somewhat wider set of conditions (consider, for example, the Gauss-Markov theorem; as long as you're not in a situation where all linear estimators are bad, the best linear estimator may be quite useful)

*often the resulting estimators may be relatively easy/efficient to compute or update. Sometimes (e.g. in some online situations) you may only have time to compute/update something that's very simple to compute or update.

*sometimes it may be difficult to know what else to assume, for example if sample sizes are small and you have no particularly good idea what the errors might look like 

*It may be a good starting place for something that doesn't actually assume Gaussian noise (e.g. robust procedures which are designed to perform well under contaminated normality but which perform quite well in a much wider variety of situations). This may be particularly useful when in the situation described in item 5.

Addressing the edit (though an edit above pretty much covers it):

Why do we need iid property?

In general, we don't need it. Points 1, 4, 5 and to some extent 2 above suggest some reasons why it might be used, but we can deal with both dependence and non-identical distributions if we know what form of dependence we mean and what distributions we have.
I don't quite follow what your part 3 is trying to ask. Independence is not a property of a univariate distribution; and if they're not identically distributed both cdf and density (assuming continuous r.v.s) will be different.
