Sources:

1. Jeseph K. Blitzstein & Jessica Hwang. (2019). Conditional propability. Introduction to Probability (2nd ed., pp. 369-375). CRC Press.

Change of Random Variables

Notation

Symbol	Type	Description
$X$	Random variable	Original continuous random variable
$f_X(x)$	Function	Probability density function (PDF) of $X$
$Y$	Random variable	Transformed continuous random variable
$f_Y(y)$	Function	Probability density function (PDF) of $Y$
$g(x)$	Function	Transform function, assumed to be differentiable and strictly monotonic
$g^{-1}(y)$	Function	Inverse of the transformation function $g(x)$
$\mathbf{X}$	Vector	Original random vector $(X_1,X_2,\dots ,X_n)$
$f_{\mathbf{X}}(\mathbf{x})$	Function	Joint PDF of $\mathbf{X}$
$\mathbf{Y}$	Vector	Transformed random vector $(Y_1,Y_2,\dots ,Y_n)$
$f_{\mathbf{Y}}(\mathbf{y})$	Function	Joint PDF of $\mathbf{Y}$
$\frac{\partial \mathbf{x}}{\partial \mathbf{y}}$	Matrix	Jacobian matrix of partial derivatives of $g^{-1}$
$det\left(\frac{\partial \mathbf{x}}{\partial \mathbf{y}}\right)$	Scalar	Determinant of the Jacobian matrix
$A_0,B_0$	Sets	Open subsets of ${\mathbb{R}}^n$, representing domain and range of $g$, respectively

Abbreviations

Abbreviation	Description
PDF	Probability Density Function
CDF	Cumulative Distribution Function
r.v.	Random variable
Jacobian	Matrix of partial derivatives

Change of variables in one dimension

Theorem: Change of variables in one dimension Let $X$ be a continuous r.v. with PDF $f_X$, and let $Y=g(X)$, where $g$ is differentiable and strictly increasing (or strictly decreasing). Then the PDF of $Y$ is given by $$f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|,$$

where $x=g^{-1}(y)$. The support of $Y$ is all $g(x)$ with $x$ in the support of $X$.

Proof:

Let $g$ be strictly increasing. The CDF of $Y$ is $$F_Y(y)=P(Y\le y)=P(g(X)\le y)=P(X\le g^{-1}(y))=F_X(g^{-1}(y))=F_X(x),$$ so by the chain rule, the PDF of $Y$ is $$f_Y(y)=\frac{\partial F_Y(y)}{\partial y}=\frac{\partial F_X(x)}{\partial y}=\frac{\partial F_X(x)}{\partial x}\frac{\partial x}{\partial y}=f_X(x)\frac{dx}{dy}$$ The proof for $g$ strictly decreasing is analogous. In that case the PDF ends up as $-f_X(x)\frac{dx}{dy}$, which is nonnegative since $\frac{dx}{dy}<0$ if $g$ is strictly decreasing. Using $\left|\frac{dx}{dy}\right|$, as in the statement of the theorem, covers both cases.

Change of variables in multiple dimensions

Theorem: Let $\mathbf{X}=(X_1,\dots ,X_n)$ be a continuous random vector with joint PDF $f_{\mathbf{X}}(\mathbf{x})$. Let $g:A_0\to B_0$ be an invertible function, where $A_0$ and $B_0$ are open subsets of ${\mathbb{R}}^n$. Suppose $A_0$ contains the support of $\mathbf{X}$, and $B_0$ is the range of $g$.

Define $\mathbf{Y}=g(\mathbf{X})$, with $\mathbf{y}=g(\mathbf{x})$. Since $g$ is invertible, we have:

$$\mathbf{X}=g^{-1}(\mathbf{Y})\text{ and }\mathbf{x}=g^{-1}(\mathbf{y})$$

Assuming all partial derivatives $\frac{\partial x_i}{\partial y_j}$ exist and are continuous, the Jacobian matrix of the transformation is:

$$\frac{\partial \mathbf{x}}{\partial \mathbf{y}}=\left[\begin{array}{cccc}\frac{\partial x_1}{\partial y_1}& \frac{\partial x_1}{\partial y_2}& \cdots & \frac{\partial x_1}{\partial y_n}\\ ⋮& & & ⋮\\ \frac{\partial x_n}{\partial y_1}& \frac{\partial x_n}{\partial y_2}& \cdots & \frac{\partial x_n}{\partial y_n}\end{array}\right]$$

If $\det \left(\frac{\partial \mathrm{x}}{\partial \mathrm{y}}\right)\ne 0$ everywhere, the joint PDF of $\mathbf{Y}$ is:

$$f_{\mathbf{Y}}(\mathbf{y})=f_{\mathbf{X}}(g^{-1}(\mathbf{y}))\cdot \left|\det \left(\frac{\partial \mathbf{x}}{\partial \mathbf{y}}\right)\right|.$$

This holds for all $\mathbf{y}\in B_0$.