Change of Variable (Probability)

Univariate Case §

thm. Linear Change of Variable. let $X\text{s.t.}\mathbb{P}(\mathbb{X}=x)=f_X(x)$. let $Y=aX+b$. Then…:

$$f_Y(y)=\frac{1}{|a|}{f}_X(\frac{y-b}{a})$$

⭐ thm. Change of Variable (bijective function). let $X\text{s.t.}\mathbb{P}(\mathbb{X}=x)=f_X(x)$. let $Y=g(X)$ where $g(x)$ is an inversible function (=bijective). Then:

$$f_Y(y)=\frac{f_X[g^{-1}(y)]}{|dy/dx|}$$

• The bottom is a x-axis reflective version of $f_X$
• The left is a 90 degrees counterclockwise rotation of $f_Y$
• The graph is of $Y=g(X)$ where $g(x)=\sqrt{x}$

thm. Change of Variable (injective function). let:

• $X,Y,f_X,f_Y,Y=g(X)$
• $\text{Domain}(X)$ is partitioned $[x_1,x_2),[x_2,x_3),\dots$ s.t. $g(x)$ is bijective in each interval.

then $f_Y(y)$ can be defined on interval $[x_i,x_{i+1})$…:

$$f_Y(y)=\sum _{x:y=g(x)}\frac{f_X[g^{-1}(y)]}{|dy/dx|}\text{ , when }y\in g([x_i,x_{i+1}))$$

Multivariate Case §

thm. Change of Variables. (bijective function). let $\vec{X}$ with $f_{\vec{X}}(\vec{x})$ and $\vec{Y}=\mathbf{g}(\vec{X})$, where $\mathbf{g}:{\mathbb{R}}^n\mapsto {\mathbb{R}}^n$ is a bijective function with continuous derivatives. Then the joint probability density function of $Y=(Y_1,Y_2,\dots ,Y_n{)}^{\top }$ is:

$$f_{\mathbf{Y}}(\mathbf{y})=\frac{f_{\mathbf{X}}({\mathbf{g}}^{-1}(\mathbf{y}))}{|\det \underset{\text{ jacobian }}{\underbrace{\frac{\partial \mathbf{x}}{\partial \mathbf{y}}}}|}$$

Example. See the Box-Mueller Transform