Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sample distribution theory using Coarea Formula   Let $\left(\Omega,\Sigma,p\right)$ be a probability measure space and let
$X:\Omega\to{\mathbb{R}}^k$ be a (vector valued) random variable. We suppose
that the probability $p_X$ induced by $X$ is absolutely continuous with respect
to the Lebesgue measure on ${\mathbb{R}}^k$ and set $f_X$ as its density
function. Let $\phi:{\mathbb{R}}^k\to {\mathbb{R}}^n$ be a $C^1$-map and let us
consider the new random variable $Y=\phi(X):\Omega\to{\mathbb{R}}^n$. Setting
$m:=\max\{\mbox{rank }(J\phi(x)):x\in{\mathbb{R}}^k\}$, we prove that the
probability $p_Y$ induced by $Y$ has a density function $f_Y$ with respect to
the Hausdorff measure ${\mathcal{H}}^m$ on $\phi({\mathbb{R}}^k)$ which
satisfies \begin{align*} f_Y(y)=
\int_{\phi^{-1}(y)}f_X(x)\frac{1}{J_m\phi(x)}\,d{\mathcal{H}}^{k-m}(x), &\quad
\text{for ${\mathcal{H}}^m$-a.e.}\quad y\in\phi({\mathbb{R}}^k). \end{align*}
Here $J_m\phi$ is the $m$-dimensional Jacobian of $\phi$. When $J\phi$ has
maximum rank we allow the map $\phi$ to be only locally Lipschitz. We also
consider the case of $X$ having probability concentrated on some
$m$-dimensional sub-manifold $E\subseteq{\mathbb{R}}^k$ and provide, besides,
several examples including algebra of random variables, order statistics,
degenerate normal distributions, Chi-squared and "Student's t" distributions.

26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio