Push-forward density as surface integral [closed]

Question

Let $X$ be a random variable taking values in $\mathbb R^n$ with a probability distribution $\mathbb P$ that has a density $p$.

Consider further a linear mapping $\pi: \mathbb R^n \to \mathbb R^m$, i.e. $\pi$ is an $m \times n$ matrix. We assume $m<n$, i.e. the linear transformation is in general non-invertible!

Now the distribution of the new random variable given by $Y = \pi X$ is given by

$$ \mathbb P_Y (B \subset \mathbb R^m) := \mathbb P(\pi^{-1}(B)) = \int_{\pi^{-1}(B)} p \; dx$$

where $\pi^{-1}(B)$ denotes the pre-image of $B$ under $\pi$.

Question: How does the density of the random variable $Y$ look like? Is it simply:

$$ p_Y(y) = \int_{\pi^{-1}(\{y\})} p \; dS$$

or do I need some scaling?

Initially I had the following:

$$ \int_{B} p_Y(y) \; dy = \int_{B} \int_{\pi^{-1}(\{y\})} p \; dS \, dy = \int_{\pi^{-1}(B)} p \; dx = \mathbb P_Y(B),$$

but there seems some scaling missing. Could you please tell me where my mistake is? Thank you

Answer 1

First of all, let us assume that $\pi$ is onto to avoid some unnecessary complications. Then we can find linear coordinates $x=(x_1,\dotsc, x_n)$ on $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$ and $y=(y_1,\dotsc, y_m)$ on $\bR^m$ such that, in these coordinates, $\pi$ is given by

$$ y_j=x_j,.\;\;\forall j=1,\dotsc, m. $$

The probability measure $\newcommand{\bP}{\mathbb{P}}$ $\bP$ then has the form $$ \bP(dx)= \rho(x) dx. $$

Then

$$\bP_Y(dy)= \rho_Y(y) dy,\;\;\;\rho_Y(y_1,\dotsc,y_m)=\int \rho(y_1,\dotsc, y_m, x_{m+1}, \dotsc x_n) dx_{m+1}\cdots dx_n. $$

If $x$ and $y$ are arbitrary orthonormal coordinates on $\bR^n$ and resp. $\bR^m$, then $\pi$ is represented by a matrix $A$ and we set

$$J_\pi:=\sqrt{\det AA^*}. $$

This quantity $J_\pi$, called the Jacobian of of $\pi$ is independent of the choice of orthonormal coordinates $x$ and $y$.

If $B\subset \bR^m$ is a Borel set, then the coarea formula

$$ \int_{\pi^{-1}(B)} \rho(x) dx = \int_B \;\underbrace{\left(\int_{\pi^{-1}(y)}\frac{\rho(x)}{J_\pi} dV_{\pi^{-1}(y)} (x)\right)}_{=:\rho_Y(y)}\; dy, $$

where $dV_{\pi^{-1}(y)}$ denotes the volume element on the affine plane $\pi^{-1}(y)$ induced by the Euclidean metric on $\bR^n$. For more details check the first two pages of these notes.