### Question

**What is the probability distribution function (PDF) of the absolute volume of a tetrahedron with random coordinates?**

The 4 random tetrahedron vertices in $\mathbb{R}^3$ are $$ \mathbf{\mathrm{X}_1} =(x_1^1,x_1^2,x_1^3),\;\; \mathbf{\mathrm{X}_2}=(x_2^1,x_2^2,x_2^3),\;\; \mathbf{\mathrm{X}_3}=(x_3^1,x_3^2,x_3^3),\;\; \mathbf{\mathrm{X}_4} =(x_4^1,x_4^2,x_4^3)$$ where $x_i^j$ are independent standard normal distributed variables $$x_i^j\sim\mathcal{N}(0,1)$$ The non-oriented volume of a random tetrahedron instance is $$V=\frac{1}{6}\left| ( \mathbf{\mathrm{X}_1}- \mathbf{\mathrm{X}_4})\cdot \left(( \mathbf{\mathrm{X}_2}- \mathbf{\mathrm{X}_4} ) \times ( \mathbf{\mathrm{X}_3}- \mathbf{\mathrm{X}_4} )\right) \right| \tag{1} $$ $$=\frac{1}{6}\left|x_1^2 x_2^3 x_3^1 - x_1^1 x_2^3 x_3^2 + x_1^3 x_2^1 x_3^2- x_1^2 x_2^1 x_3^3 + x_1^1 x_2^2 x_3^3- x_1^3 x_2^2 x_3^1 + x_1^3 x_2^2 x_4^1- x_1^1 x_2^2 x_4^3 + x_1^1 x_2^3 x_4^2- x_1^2 x_2^3 x_4^1 + x_1^2 x_2^1 x_4^3- x_1^3 x_2^1 x_4^2 + x_2^3 x_3^2 x_4^1 -x_2^1 x_3^2 x_4^3 + x_2^1 x_3^3 x_4^2- x_2^2 x_3^3 x_4^1 + x_2^2 x_3^1 x_4^3- x_2^3 x_3^1 x_4^2 + x_1^2 x_3^3 x_4^1- x_1^1 x_3^3 x_4^2 + x_1^3 x_3^1 x_4^2- x_1^2 x_3^1 x_4^3 + x_1^1 x_3^2 x_4^3- x_1^3 x_3^2 x_4^1 \right|$$

### Known relations

The expectation value of $V$ is $$\mathbb{E}[V]=\frac{2}{3}\sqrt{\frac{2}{\pi}}\tag{2}$$ A proof can be found in a Math Stack Exchange post.

The variance of $V$ is $$\mathbb{Var}[V]=\mathbb{E}[V^2]-(\mathbb{E}[V])^2=\frac{2}{3}-\frac{8}{9\pi}\tag{3}$$ where $\mathbb{E}[V^2]$ can be calculated by multiple integration.

### Approximate relations based on empirical data

*The remaining part contains only unproven statements that could give indications of the true solution.*

The probability distribution of empirical data of $V$ can be fitted quite well with a function of the form $$f(V)=\text{exp}\left(-\left(\frac{V}{c_2}\right)^{c_1}\right)c_3\tag{4}$$ where $c_1,c_2,c_3$ are fit parameters. As a PDF must fulfill the conditions $$\int_0^\infty f(V) \mathrm{d}V=1\ \,\, \text{and}\ \int_0^\infty V f(V)\mathrm{d}V=\mathbb{E}[V]$$ the fit parameters $c_2$ and $c_3$ in eq.(4) can be expressed in dependence of $c_1$ $$c_2=\mathbb{E}[V]\frac{\Gamma(1/c_1)}{\Gamma(2/c_1)}\ ,\;\; c_3=\frac{c_1}{c_2\Gamma(1/c_1)}$$ with $\Gamma$ being the Gamma function. Only $c_1$ remains to be fitted. The best fit is for $c_1\approx\pi/4$, i.e. $c_1\approx 0.7854, c_2\approx 0.3491, c_3\approx 2.4944$. However it is not known whether eq.(4) is the true form of the PDF at all. It just models well experimental data.