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Problem

Suppose I sample a multivariate standard normal sample $x \sim N(0, I_d)$ where $d \geq 1$ is the dimension. What is the expected value of the norm of $x$ $$ \mathbb{E}[\|x\|] = ? $$ Using python, I have sampled 100k samples in $2$ and $5$ dimensions and obtained an average of $1.255$ and $2.125$ respectively. Of course this number depends on the dimensionality.

Bonus Problem

Futher question: how does this change for $x\sim N(\mu, \Sigma)$?

Solution Attempt

$$ \mathbb{E}[\|x\|] = \int_{\mathbb{R}^d} \|x\| (2\pi)^{-\frac{d}{2}}\exp\left\{-\frac{1}{2} \|x\|^2\right\} dx $$

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    $\begingroup$ $x \sim N(0, I_d)\implies\|x\|^2\sim \chi^2_d$, from which one gets $\mathbb{E}[\|x\|]=\sqrt 2\frac{\Gamma((d+1)/2)}{\Gamma(d/2)}$. See stats.stackexchange.com/q/144893/119261 and linked posts. $\endgroup$ Commented Sep 8, 2021 at 10:50

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