

Subject:
(Probability) distribution from full set of moments
Category: Science > Math Asked by: borelga List Price: $15.00 
Posted:
02 Apr 2006 21:08 PDT
Expires: 02 May 2006 21:08 PDT Question ID: 714834 
I'm looking for detailed explanation (or even proof) of how to reconstruct a distribution from full set of moments. I've found this page so far, but it doesn't contain a proof and it's also unclear about exact conditions: http://www.plmsc.psu.edu/~www/matsc597/probability/moments/node3.html#SECTION00030000000000000000 This should be fairly classical mathematical result. 

Subject:
Re: (Probability) distribution from full set of moments
Answered By: hedgiega on 02 Apr 2006 22:02 PDT Rated: 
Hi borel The 'missing link' is SEARCH TERMS: The Characteristic Function of a Probability Distribution Its Taylor series gives you moments The Characteristic Function as a MomentGenerating Function it's Fourier tranfrom is the distribution http://www2.sjsu.edu/faculty/watkins/charact.htm see also http://en.wikipedia.org/wiki/Characteristic_function http://mathworld.wolfram.com/UniformDistribution.html cumulant http://mathworld.wolfram.com/UniformDistribution.html etc. Hope this explains the onetoone correspondence Hedgie  
 

borelga
rated this answer:
Although these links are not exactly what I was hoping for, they have pointed me to the right direction. The detailed answer I hoped for is obviously only in textbooks and not on the internet. Thank you Hedgie. 

Subject:
Re: (Probability) distribution from full set of moments
From: mathisfunga on 03 Apr 2006 06:28 PDT 
Well I'm guessing you will have a hard time finding the answer to this question in a formal proof at least. My final stats course book (in a stats major) gives the theorem that if the moment generating functions for distributions X and Y are equal (mx(t)=my(t)) for all values of t then X and Y have the same prob. dist. However it adds "the proof of this theorem is beyond the scope of this text" Now having all the moments to a function seems like a hard task in most cases for me, as many cases there are infinite moments. However having the m.g.f. in closed form allows you of course to generate any moment you want. Thusly if the mgfs are equal all moments must be equal, from this we know that E{Y^k)=E(X^k) for all k such that k is in Z+ Where to go from there is where I get lost, obviously intuition leads you to Y=X but intuition doesn't amount to much. 
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