I would like to calculate the mean and variance of the normed distance of a cone-shaped distribution,

$f(x) \propto \exp(-|x|)$,

where $x\in\mathbb{R}^d$, where $d$ can be any positive integer.

In two-dimensions, this distribution looks like

plot

a cone! I can calculate the normalising constant for this distribution using,

Integrate[Exp[-Norm[{x,y}]], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]

which is $2\pi$. I can then calculate the mean normed distance using,

Integrate[Norm[{x,y}]Exp[-Norm[{x,y}]], {x, -Infinity, Infinity},
                                        {y, -Infinity, Infinity}]

which is 2. Its second moment,

Integrate[Norm[{x,y}]^2Exp[-Norm[{x,y}]], {x, -Infinity, Infinity},
                                          {y, -Infinity, Infinity}]

then allows me to calculate the variance $\mathrm{Var}(|x|) = \mathrm{E}(|x|^2) -\mathrm{E}(|x|)^2 = 2$.

Calculating the normalising constants is easy enough in higher dimensions, but I run into trouble with finding the mean and variance.

Any ideas?

I'm guessing that something can maybe be done using polar coordinates in higher dimensions but this isn't something I know much about!

  • 4
    Perhaps you could edit your post to clarify that you're not after the mean and variance, but the mean normed distance and its variance? – Chris K Sep 25 at 0:35
  • Good idea. I have now added this to the title and the description. – ben18785 Sep 25 at 22:39
up vote 10 down vote accepted

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]

$$n$$

  • Thanks! Just a small thing, the expectation shouldn't be zero because I am calculating the expectation of the normed x, not x itself. – ben18785 Sep 25 at 0:00
  • You're welcome. – Henrik Schumacher Sep 25 at 0:05
  • Sorry, just noticed that the variance here isn't quite right (it's actually the second moment). It needs to have the mean-squared subtracted from it. Think this is because we first said the mean was zero. Anyway, may be worth mentioning in the answer somewhere. – ben18785 Sep 25 at 22:18
  • 1
    Yeah, good point. Actually, I already realized that this morning but did not find time to correct it. – Henrik Schumacher Sep 25 at 22:50
  • 1
    I fixed also a typo in the surface area of the spheres. (I did not change the results in the end because we normalize anyway.) – Henrik Schumacher Sep 25 at 22:56
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}

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