Change-of-Variable Formula (Jacobian Matrix)

Given a random variable $z$ and its known probability density function $z\sim \pi (z)$, we would like to construct a new random variable using 1-1 mapping function $x=f(z)$. The function $f$ is invertible, so $z=f^{-1}(x)$. Now the question is how to infer the unknown probability density function of the new variable, $p(x)$?

$$\int p(x)dx=\int \pi (z)dz=1\text{ , Definition of prob. distr.}$$ $$p(x)=\pi (z)\left|\frac{dz}{dx}\right|=\pi (f^{-1}(x))\left|\frac{d{f}^{-1}}{dx}\right|=\pi (f^{-1}(x))|(f^{-1}{)}^{\prime }(x)|$$

The Change-of-Variable Formula

Suppose that the variables $x$ and $y$ are related to the variables $u$ and $v$ by the equations $x=x(u,v),y=y(u,v)$. Then

${\iint }_{R_{xy}}f(x,y)dxdy={\iint }_{R_{uv}}f[x(u,v),y(u,v)]\left|\frac{\partial (x,y)}{\partial (u,v)}\right|dudv$

$\frac{\partial (x,y)}{\partial (u,v)}=\det \left[\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}\end{array}\right]$

This function is called the Jacobian of the transformation, denoted by $J$.

The Jacobian is a factor that is introduced to compensate for the distortion of the domain that occurs when we move it from one coordinate system to another.

Restriction to formula

There is one important restriction on the transformation $(x,y)\to (u,v)$; we must not have $J=0$ at any point on the interior of $R_{uv}$​. This condition ensures that the transformation is invertible on the domain of integration.

This is where the Jacobian comes from when moving from one coordinate system to another. $|(f^{-1}{)}^{\prime }(x)|$ indicates the ratio between the area of rectangles defined in two different coordinate of variables $z$ and $x$ respectively.