# probability contained in a differential area must be invariant under change of variables

The Wiki article on PDFs describes the transformations of random variables.

In deriving the formula (one-dimensional), the article states "This follows from the fact that the probability contained in a differential area must be invariant under change of variables."

$$|f_Y(y)dy|=|f_X(x)dx|$$

I do not understand what "probability contained in a differential area" means and why it must be invariant under transformation.

Wikipedia: Change of Variable

It means that if transformation is $$Y=g(X)$$, we should have $$P(X=x)=P(g(X)=g(x))=P(Y=g(x))=P(Y=y)$$ where, $$g(x)=y$$, where $$g$$ is monotonic.
Since these are continuous random variables, $$P(X=x)$$ is normally $$0$$, because it's the limit $$P(X=x) = \lim_{dx\rightarrow 0} f_X(x)|dx|$$
i.e. the integral under the curve while the limit goes to $$0$$. However, at infinitesimal scale, the probabilities, i.e. $$P(x and $$P(y must be equal. These are $$f_X(x)|dx|$$ and $$f_Y(y)|dy|$$ respectively.
A better way to think in this scale is to have the approximation $$P(X=x)\approx f_X(x)|dx|$$.
• Because we're interested in the area/length of the infinitesimal region. For example, if $Y=-X$, then $dy=-dx$. Probabilities can't be negative, we need to remove the sign. Dec 16, 2020 at 5:21