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Given a random variable $X$, consider the k-th (non-central) moment about $a$, $E \left[ ( X - a )^k \right]$

Is there any relation of this value to the chracteristic function of variable's probability distribution like there is for the raw moments? i.e. do we have any identity similar to this one? $$E[X^k] = (-i)^k \varphi_X^{(k)}(0)$$ where $\varphi_X(\cdot)$ is the characteristic function of the random variable $X$

I mean if I know the characteristic function, calculating the non central moments should be just a simple application of the binomial theorem(or am I missing something?), but what I am really after is an understanding about what concrete effects the rest of the characteristic function has on the distribution. This identity makes it almost seem like the only important point it at zero.

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Recall that $\varphi_X(t) = E[e^{itX}]$. Hence, $\varphi_{X-a}(t) = E[e^{it(X-a)}] = e^{-ita}\varphi_X(t)$. You can compute the moments using your formula for $\varphi_{X-a}(t)$.

Regarding your second question, I guess it corresponds to: "When is a distribution completely specified by it's moments?". An answer is here.

PS: Any analytic function is completely defined by it's derivatives on one point! So, if the characteristic function is analytic, the derivatives on $0$ have all the information about the characteristic function and the characteristic function has all the information about the distribution!

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Aren't you missing the expectation in the definition of phi? Also, not every characteristic function is analytic, but the characteristic function is defined for every distribution, even ones where the central moments don't exist. –  Tim Seguine Nov 2 '12 at 19:50
    
You are right! The PS only makes sense when the characteristic funcion is analytic. –  madprob Nov 2 '12 at 19:58
    
That post is interesting as well, but my second question since it was a little vague, I will try to rephrase. Take for example the Cauchy distribution; it has no well defined moments. What information can we gain then from it's characteristic function that might not have been apparent from its PDF? One of the obvious things is that since it is not differentiable in zero, that it doesn't have any higher moments, but can we see anything else? It seems like that since the fourier transform is so useful in other fields, that there should be other insights available to us as well. –  Tim Seguine Nov 2 '12 at 20:34

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