I am trying to understand how to determine the limits of the integral for calculating the cdf of a fairly simple transformation, but struggling conceptually to understand the support.
Would it be accurate to say the integral when calculating the cdf, goes from $-\infty$ to $0$ ? If so, isn't the transformation divergent? How do you determine the cdf in such cases? Does the cdf just not exist?
Assuming that $X \in \mathbb R$, that is to say, the support of $X$ is the set of all real numbers, then the support of $Y$ is found by considering how $$Y = \begin{cases} X^2, & X \ge 0 \\ -2X, & X < 0 \end{cases}$$ maps each of the intervals $[0, \infty)$ and $(-\infty, 0)$ under this transformation. In the first case, clearly $X^2 \ge 0$ when $X \ge 0$, thus $Y \ge 0$. In the second, when $X < 0$, then $-2X > 0$, consequently, $Y > 0$. It follows that the support of $Y$ is the interval $[0, \infty)$.