I've just read about probability density function from this article.
In that article, there is some wired concept that I can't understand, please see the section named "Dependent variables and change of variables".
In that section, there is an equation appeared with short comment about it, "This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is, (and the equation appeared)"
I can't understand what that equation means and the sentence "the probability contained in a differential area must be invariant under change of variable", especially the "invariant under change of variable". Why the probability must be invariant under change of variable? and what is the change of variable?
Could anybody give me some insight to understand that equation?
The wording is kind of throwing me off, but I think I get.
The probability of $x$ being in the infinitesimal area $dx$ is $f_X(x)dx$, because the first term $f_X$ is the probability density and the second term $dx$ is the surface of the infinitesimal area. If $g(x)$ maps values of the variable $x$ in $dx$ to values in the area $dy$, then the possibility of $y$ being in $dy$ would have to be equal to the possibility of $x$ being in $dx$. Hence
$$ f_Y(y)dy=f_X(x)dx $$
Note that the probability is expressed as a product $f_X(x)dx$ only because $dx$ is considered "small enough" for the value of $f_X$ to be considered a constant. If the area was not infinitesimal, you would have to integrate $f_X(x)dx$ over it.