Hi, Any help with this appreciated.

If I have two column vectors $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$, where the $\displaystyle n$ elements of each are drawn independently from a Guassian $\displaystyle \mathcal{N}( \mu, \sigma^2)$, then the distance, $\displaystyle d$ ,between them is given by

$\displaystyle d = \sqrt{(\mathbf{a} - \mathbf{b})^\mathsf{T}(\mathbf{a} - \mathbf{b})}$

But what is the expected value of $\displaystyle d$?

I reckon, in the case where $\displaystyle \mu = \vec{0}$, it's $\displaystyle \mathbb{E}[d] = \sqrt{2n\sigma^2}$

My feeling is that a non-zero $\displaystyle \mu $ should make no difference, as it's just shifting the origin, so to speak, and the distance is determined by the relative positions of the vectors.

Can anyone prove the the general case (for$\displaystyle \mathcal{N}( \mu, \sigma^2)$ ) or show it to be wrong, and if wrong, say what it is in fact in the general case?

I notice also that my expression bears resemblance to the denominator in the normalising term in the Gaussian pdf $\displaystyle 1/\sqrt{2 \pi \sigma^2}$, except that n takes the place of $\displaystyle \pi$. Is this coincidence, or does it reflect something deeper?

Thanks in advance. MD