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I would like to find a (closed nice) expression for the non-centered Gaussian moments with mean $\mu$ and variance $\sigma$. In found something in wikipedia:

http://en.wikipedia.org/wiki/Normal_distribution#Moments

$$E|X|^p = \sigma^p 2^{p/2} \frac{\Gamma \left(\frac{p+1}{2}\right)}{\sqrt{\pi}} {}_1 F_1 \left(-\frac{1}{2}p, \frac{1}{2}, -\frac{1}{2}(\mu/\sigma)^2\right)$$

but I don't understand this "confluent hypergeometric function" ${}_1F_1$ because it doesn't seem to be well-defined for negative integers according to this article in wikipedia.. (it seems to be only valid for positive integers)

http://en.wikipedia.org/wiki/Confluent_hypergeometric_function

Does anyone know about this? Thank you very much for the help!

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1 Answer 1

Suppose $Z\sim N(0,1)$ and $X=\mu+\sigma Z$, so that $X\sim N(\mu,\sigma^2)$.

$$ \mathbb E(X^n) = \mathbb E((\mu+\sigma Z)^n) = \sum_{k=0}^n \binom n k \mu^{n-k} \sigma^k \mathbb E(Z^k). $$

If you know the moments of $Z$, plug them in there.

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That could be one possibility yes, thanks! But I still wonder what they mean by ${}_1 F_1(a,b,c)$, it is strage.. i don't find anything related.. –  Dan Mar 17 '13 at 18:52

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