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Suppose X is a d-dimensional random vector where $X \sim N(\mu, \sigma^2 I_d)$. What is $\mathbb{E}[\exp(-\alpha \|X\|)]$ ,where $\alpha>0$?

Since norm is not squared, it can't be factorized and we can't take advantage of independence. Otherwise, it was easy to solve. Any help is appreciated.

mathlover
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  • Is $μ$ a vector or a scalar? – Kodiologist Sep 09 '17 at 18:54
  • @Kodiologist It's a d-dimensional vector. – mathlover Sep 09 '17 at 19:16
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    even in the simplest case where $\mu = 0$ and $\sigma^2=1$ the result is the [MGF of a chi distribution](https://en.wikipedia.org/wiki/Chi_distribution#Moment_generating_function) which is not particularly nice. What kind of answer are you expecting? Do you expect there to be a nice form? Based on that simple case, even an answer in terms of friendly special functions may be very complicated – jld Sep 09 '17 at 19:33
  • @Thanks Chaconne. That's good to know. No, I don't expect it to be nice. I thought it might be a known result. – mathlover Sep 09 '17 at 20:35
  • @Chaconne I checked out and it seems that $\|X\|^2$ has chi-square distribution. Is it still the case with $\|X\|$? Cause I'm only interested in $\|X\|$. – mathlover Sep 11 '17 at 00:56
  • If $\mu \neq 0$ then $\frac 1{\sigma^2}||X||^2 \sim \chi^2_d(\delta)$ where $\delta = \mu^T\mu$ (so it's a non-central chi-squared). – jld Sep 11 '17 at 01:28
  • Maybe ideas from https://stats.stackexchange.com/questions/318220/distribution-of-the-magnitude-of-samples-from-the-multivariate-gaussian-distribu/318918#318918 can be used (and extended) – kjetil b halvorsen Oct 08 '18 at 14:19

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