Why Cant We Directly Find the PDF of the Transformation of Random?

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Why can't we directly find the PDF of the transformation of random variables, say g(X) from the random variable X. Why do we have to first convert PDF into CDF and then have to differentiate to get to the PDF of the transformed random variable?

Let (\Omega,\mathcal{F},\mathbb{P}) be the probability space of interest. It is given that the random variable X\sim \text{Unif}(0,1). The random variable Y is defined as Y(\omega).=\begin{cases}X(\omega) & \quad\text{ if }00\ge y for all \omega\in\Omega. Thus, \mathbb{P}(Y\le y)=0. Now, let 0

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Let F_y = 2/by. This CDF is given by F_P(y)=2/x2. Now, since there exist random variables P(0,1) and P(0,1/2) such that P(1/2

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