Why Is the Characteristic Function of a Random Variable Complex-Valued?

[cappadonza]

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 don't 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 ?

 

[mathman]

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.

 

[statdad]

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]

I found this presentation to be helpful in understanding the derivation and applications of the characteristic function.

http://www.sjsu.edu/faculty/watkins/charact.htm