I have some (analytical) function $y=f(x)$, where $x,y\in\mathbb{R}$. I sample this function at equidistant points $x_i = i\Delta x$, where $\Delta x\in\mathbb R$ and $i\in\mathbb N$. This gives a sequence $\{y_i\}_{i\in\mathbb N}$. Then I compute the empirical probability distribution function for $\{y_i\}_{i\in\mathbb N}$.
I am interested to know whether one can derive the probability distribution function for $\{y_i\}_{i\in\mathbb N}$ of from $f(x)$ or any of its properties ---- especially in the case $\Delta x\rightarrow 0$. Is this possible general? Or for a certain type of function?
It seems obvious that for $y=mx+n$ with $m,n\in\mathbb R$, one would get the uniform distribution over $\mathbb R$. But I do not know what kind of math I need to extend this to any other type of function.
Let $f(x)$ be monotonically increasing and continuous over a finite support $[A,B]$. And let $X$ be a random variable uniformly distributed over $[A,B]$. Then the distribution function sought for can be written as $$P(f(X)<x)=P(X<f^{-1}(x))=\frac{f^{-1}(x)}{B-A}.$$
If $g$ is monotonically deceasing then
$$P(f(X)<x)=P(X>f^{-1}(x))=1-\frac{f^{-1}(x)}{B-A}.$$
If $f$ is a function composed of parts like those considered above then, again, such a distribution function can be given.