I was doing some exercices that was offered during the lecture and came across very interesting one. I didn't make it, then asked my teacher and didn't understand what He tried to explain me. So here it is:
I have to explain the probability distribution function $F_\nu(t)=\nu((-\infty,t])=\mu(f^{-1}(-\infty,t])$ in terms of function $F_\mu(t)$. Where $f(x)=\frac{x}{x^2+1}$.
For example if $f(x)=x^2$ then $F_\nu(t)=0$ if $t<0$ and $ F_\nu(t)=F_{\mu}(\sqrt{t}) - F_\mu(-\sqrt{t}) $ when $t\geq 0$.
I think that it would be very helpful to find an inverse function. I don't know if it exists, but I remember that during the calculus when we had to change variables and we had functions like $\sqrt{2x-x^2}$ there was an inverse function on some interval and it was given like $1+\sqrt{1-y^2}$. I don't know if it's right, but I think I need something like this in this situations.