$X \sim Unif(a, b)$, $a$ and $b$ are arbitrary numbers.
I need to calculate the density of $Y = X^2$. I know about Method of transformations. But in my case the function is not invertible on $R$. However, the function is invertible on the positive and negative half-axes. Is there any way to apply this here?
I also tried to calculate directly through the distribution function. But with any $a$ and $b$ this is difficult.
Consider the the equivalent events.
If $a,b>0$ or $a,b<0$, then $f_Y(y) = \begin{cases} \frac{1}{2|b-a|\sqrt{y}}, & a^2<y<b^2\\ 0, & \text{o.w.} \end{cases}$.
If $a<0$ and $b>0$, let $m=\min\{|a|,|b|\}$, and $M=\max\{|a|,|b|\}$, then
$f_Y(y) = \begin{cases} \frac{1}{(M+m)\sqrt{y}}, & 0<y<m^2\\ \frac{1}{2(M+m)\sqrt{y}}, & m^2<y<M^2\\ 0, & \text{o.w.} \end{cases}$
because $y \in (0,m^2)$ occurs if $x = \pm \sqrt{y}$, but $y\in(m^2, M^2)$ only occurs if $x=\sqrt{y}$, when $M=b$, or if $x=-\sqrt{y}$, when $M=|a|$.