Transformations of Random Variables

One of the fundamental problems in statistics is understanding how a random variable propagates within a complex system and to what extent it renders the output of that system random.

Let $f(\cdot)$ be a function that transforms the random variable $X$ into the random variable $Y$, that is, $y = f(x)$, with $x$ realizations of the random variable $X$, and suppose that $f$ is invertible, meaning that there exists a function $x = g(y)$ such that $g(f(x))=x$.

Let $\mathcal{I}_x$ be a generic interval of the domain of existence of the values $x$ and $\mathcal{I}_y=\{ y: y=f(x), x \in \mathcal{I}_x \}$ its corresponding image. It is evident that the probabilities of the events $x$ in $\mathcal{I}_x$ and $y$ in $\mathcal{I}_y$ must be equal, that is,

$$\int_{\mathcal{I}_y} p_Y(y) dy = \int_{\mathcal{I}_x} p_X(x) dx$$ (2.23)

Without loss of generality, it is possible to set the interval $\mathcal{I}_x$ to an infinitesimal value. Under this condition, the relationship (2.23) simplifies to

$$p_Y(y) \vert dy\vert = p_X(x) \vert dx \vert = p_X(g(y)) \vert dx \vert$$ (2.24)

from which we obtain It seems that you have entered a placeholder for a mathematical block. Please provide the specific content or equations you would like me to translate, and I will be happy to assist you! This relationship can be easily extended to the case of non-injective functions by summing the different intervals, and to the multidimensional case by using the Jacobian instead of the derivative.