 Change-of-variables formula for a probability density

Let a random vector $X$ have probability density function $f_X$ on an open set $U\subseteq {\mathbb{R}}^d$. If $g:U\to V$ is a continuously differentiable diffeomorphism onto an open set $V$, with nonsingular Jacobian matrix, then $Y=g(X)$ has probability density functionand zero off $V$. A diffeomorphism includes differentiability of the inverse; a merely differentiable bijection is not the full hypothesis needed for this density formula. The determinant corrects local volume, and its absolute value accounts for an orientation reversal.

Indeed, for a Borel set $A\subseteq V$, apply the change of variables formula to $\mathbb{P}(Y\in A)={\int }_{g^{-1}(A)}{f}_X(x)dx$. The resulting integral over $A$ identifies the probability density function.

For example, if $X$ has an exponential distribution of rate $\lambda >0$ and $Y=X^2$, use the diffeomorphism $(0,\infty )\to (0,\infty )$. It gives $f_Y(y)=\lambda e^{-\lambda \sqrt{y}}/(2\sqrt{y})$ for $y>0$. Its integrable singularity at zero does not create a point mass. A map with several inverse branches requires summing their density contributions rather than choosing a single inverse.