# Looking for a probability distribution in $\mathbb{R}^n$: exponentially decreasing density as distance from point

I was thinking of a probability distribution over points in $\mathbb{R}^n$ of the following form. Let $\mu \in \mathbb{R}^n$, $b$ a scale parameter, and $X$ a random variable in $\mathbb{R}^n$; then $\Pr[X = x] \propto e^{-\|x - \mu\|/b}$ where $\|a\|$ is the $L_2$ norm of $a$.

The motivation is to generalize the Laplace distribution to $\mathbb{R}^n$.

I think the density function should look something like

$$\Pr[X=x] = \frac{1}{b^n S_{n-1}} e^{-\|x - \mu\| / b}$$

where $S_{n-1}$ is the surface area of the $n$-dimensional unit sphere. This would give $$\Pr[X=x] = \frac{\Gamma\left(\frac{n}{2}\right)}{2\pi^{\frac{n}{2}}b^n} e^{-\|x-\mu\|/b}$$ Does this look like the right thing? Is there a name for this distribution?

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There are a few interpretations of a multivariate Laplace distribution. One is given here , and some alternatives are given here.

Also see this discussion.

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Thanks, I was missing the keyword 'multivariate' which could be very helpful! I cannot tell yet if any of the distributions mentioned in your links are equivalent to the one I'm looking for (since they don't give much geometric intuition and mainly just give the characteristic function), so any thoughts on that would be very helpful! – usul Apr 20 '13 at 1:53