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Characteristic funtion of RV

  1. Mar 25, 2010 #1
    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 [tex] \phi(t) = \mathbb{E}(e^{itX})[/tex]
    how is this actually derived, is somewhere where i can find the proof ?
  2. jcsd
  3. Mar 25, 2010 #2


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    Science Advisor

    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.
  4. Mar 25, 2010 #3


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    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.
  5. Mar 27, 2010 #4
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