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For transformations of random variables, why the value of the CDF within the corresponding range keeps the same even after applying a Nonlinear Transformation?

For example. X ~ U(0, 2), the PDF of X is $f_X(x) = \frac{1}{2}$ and the CDF is $F_X(x) = \frac{x}{2}$. Let Y = $X^2$, I can finish derivative $f_Y$ and $F_y$ by myself. However, I cannot understand well why $F_X(x) = F_Y(x^2)$ is always true.

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Let $Y = \phi(X)$ where $\phi$ is invertible. Note that (since $X>0$) $$ F_Y(y) = \mathbb{P}[Y \le y] = \mathbb{P}\left[\phi(X) \le y\right] = \mathbb{P}\left[X \le \phi^{-1}(y)\right] = F_X\left(\phi^{-1}(y)\right) $$ and you are suggesting to consider $y = \phi(x)$, so $$ F_Y(\phi(x)) = F_X\left(\phi^{-1}(\phi(x))\right) = F_X(x). $$

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