Calculate mean normed distance and normed variance of cone-shaped distribution in N-dimensions

For the normalization, we need to determine $\omega$ such that

$$\omega \, \int_{\mathbb{R}^{n}} \mathrm{e}^{-|x|} \, \mathrm{d}x = 1.$$

The first moment is given by

$$\omega \, \int_{\mathbb{R}^{n}} |x|^1 \, \mathrm{e}^{-|x|} \, \mathrm{d}x.$$

For the second, we have to compute

$$\omega \, \int_{\mathbb{R}^{n}} |x|^2 \, \mathrm{e}^{-|x|} \, \mathrm{d}x.$$

All these integrals are radially symmetric.

By introducing polar coordinates, we obtain $$\int_{\mathbb{R}^{n}} |x|^\alpha \, \mathrm{e}^{-|x|} \, \mathrm{d}x =\int_{S^{n-1}}\int_0^\infty \mathrm{e}^{-r} \, r^{\alpha+n-1} \, \mathrm{d} r \, \mathrm{d}S = \omega_n \int_0^\infty \mathrm{e}^{-r} \, r^{\alpha+n-1} \, \mathrm{d} r,$$

where $\omega_n = \frac{2 \pi^{n/2}}{\Gamma \left(\frac{n}{2}\right)}$ is the surface area of the unit sphere in $\mathbb{R}^n$.

Such integrals can be computed symbolically by Mathematica:

v[n_, α_] = 2 π^(n/2)/Gamma[n/2] Integrate[r^α Exp[-r] r^(n - 1), {r, 0, ∞}, 
  Assumptions -> α + n > 0]

$$\frac{2 \pi ^{n/2} \Gamma (n+\alpha )}{\Gamma \left(\frac{n}{2}\right)}$$

So, the $k$-th moment should equal

moment[n_, k_] = FullSimplify[ v[n, k]/v[n, 0], n ∈ Integers && n > 0]

$$\frac{\Gamma (k+n)}{\Gamma (n)}$$

which, for simplicity, equals

moment[n_, k_] = (n + k - 1)!/(n - 1)!

$$\frac{(n+k-1)!}{(n-1)!} $$

So the variance of the distance is given by

var[n_] = FullSimplify[moment[n, 2] - moment[n, 1]^2]

$$n$$

ClearAll[aa, nc, dist]
aa[n_Integer] := Array[x, {n}];
nc[n_Integer] := 2^n*Pi^((n - 1)/2)*Gamma[(1 + n)/2]
dist[n_Integer] :=  ProbabilityDistribution[Exp[-Norm[aa[n]]]/nc[n], 
  ## & @@ Thread[{aa[n], -∞, ∞}]]

Expectation[Norm[aa[#]], Distributed[aa[#], dist[#]]] & /@ Range[2, 5]

{2, 3, 4, 5}

Expectation[Norm[aa[#]]^2, Distributed[aa[#], dist[#]]] & /@ Range[2, 5]

{6, 12, 20, 30}

Also:

td[n_Integer] := TransformedDistribution[Norm[aa[n]], Distributed[aa[n], dist[n]]]

Mean[td[#]] & /@ Range[2, 5]

{2, 3, 4, 5}

Moment[td[#], 2] & /@ Range[2, 5]

{6, 12, 20, 30}

Variance[td[#]] & /@ Range[2, 5]

{2, 3, 4, 5}