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

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## Easy-to-use PDF software     ## 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?

I presume you mean Y=X^2? Let’s do it from the first principles. X\thicksim\frac{e^{\frac{-x^2}{2}}}{\sqrt{2\pi}} \mathbb{P}(Y\leq y)=\mathbb{P}(-\sqrt{y}\leq X\leq \sqrt{y})=2\mathbb{P}(0\leq X\leq \sqrt{y}) =\sqrt{\frac{2}{\pi}}\int_0^{\sqrt{y}}e^{\frac{-x^2}{2}}dx So Y\thicksim\frac{d\mathbb{P}(Y\leq y)}{dy} =\frac{d\mathbb{P}(Y\leq y)}{d\sqrt{y}}\frac{d\sqrt{y}}{dy} =\sqrt{\frac{2}{\pi}}e^{\frac{-y}{2}}\frac{1}{2 \sqrt{y}} =\frac{e^{-\frac{y}{2}}}{\sqrt{2\pi y}}.

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