# 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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