A small probability problem that I am struggling with...
Let $X \sim U[-2 , 2]$. Find the distribution of $Y = X^3 + 6$.
My main problem is the domain of $Y$.
I found that the domain of $Y$ is $-2 \leq Y \leq 14$, but I believe that the correct one is $6 \leq Y \leq 14$. Also I am a bit confused now, is that $6 \leq Y \leq 14$ actually the domain of $y$ and not $Y$?
Should I rather say that: $Y$'s domain is $[-2,14]$
So $F_Y(y) = P(Y \leq y) = P(x^3+6 \leq y) = P(x \leq \sqrt[3]{y-6})$
So actually $y$'s domain is $[6,14]$?
And then, what integrals should I take? Using what domain?
Thanks a lot!
Thanks all, i think it is sufficiently answered the question.