# Characteristic funtion of RV

so a charateristic function of a RV is complex valued funtion. from my lecture, the distribution funtion of a Random variable is not always "well behaved", may not have a density etc. A charateristic function on the other had is "well behave".
What i dont understand is, is that the only reason we use it ?
how is it actually derived, why does it have to be complex valued .
this is the definition i'm given $$\phi(t) = \mathbb{E}(e^{itX})$$
how is this actually derived, is somewhere where i can find the proof ?

You need to clarify your question. Definitions aren't derived.

As far as usage, the simplest example is deriving the distribution function a sum of independent random variables. The characteristic function of the sum is the product of the characteristic functions of the individual variables.

Homework Helper
if you are studying characteristic functions you should have already seen moment-generating functions.

moment generating functions can be used to uniquely identify the form of a distribution IF (big if) the moments of the distribution satisfy a very strict requirement. that doesn't happen all the time.

even worse, not every probability distribution has a moment-generating function: think of a t-distribution with 5 degrees of freedom: no moments of order 4 or greater, so no moment generating function.

however, EVERY distribution has a characteristic function, and every distribution is uniquely determined by the form of that function. that is one (not the only) reason for their importance.

SW VandeCarr