I'm very new in the Statistic Math field, so this question maybe be a bit trivial for you guys. Anyway, I'd appreciate any guidance in this matter.
I was thinking about whether is possible to find the distribution for a function $f(x_i)$, with random variable $x_i$ satisfying $-1 \lt x_i \lt 1$, and finite $i=1,2,...,N$. For instance, suppose we have
\begin{equation} f(x_i) = \frac{x_i}{|x_{i}|^3}, \end{equation}
Would be possible to find analytically which distribution function the histogram for $f(x_i)$ follows, for finite N? If it's not, can I find the distribution for $N\rightarrow \infty$?
I should also say that I have tried to implement this problem numerically, here is the result I'm getting https://drive.google.com/file/d/1G4ym5Afw8F9c17prd0daqpZCUuOtr1Xv/view?usp=sharing
We clearly can see that this doesn't follow the normal distribution neither other common distributions at first sight. Also, the middle gap region is a feature attributed to the local minimum of $f(x_i)$, due to the constraint of the $x_i$ value.
It depends on what you mean by "find the distribution of." We can generally find $P(f(x_i) \le c)$ for most functions and random variables, so in that sense yes. But in general it will not be any named distribution like the normal distribution. It is very rare that functions (besides linear transforms) of random variables with well-known distributions also have well-known distributions because it is often easier to just work with the original distribution of $x_i$ for computing things like the mean or variance.