Find the density of the random variable $Y=X^2$

Question

Find the density of the random variable $Y = X^2$, if the random variable $X$ follows a standard normal distribution.

I think I should use mgf to solve it is that right ? what should I do to start ?

Answer 1

This $might$ do it... We have that $$F_{X^2}(x) = \textbf{P}(X^2 \leq x) = \textbf{P}(-\sqrt{x} \leq X \leq \sqrt{x}) = 2\int_0^{\sqrt{x}}\frac{e^{-x'^2/2}}{\sqrt{2\pi}}dx'$$. Since $f_{X^2}(x) = dF_{X^2}(x)/dx$ it follows that $$ f_{X^2}(x) = \left(\frac{d}{dx}\sqrt{x}\right)\sqrt{\frac{2}{\pi}}e^{-x/2} = \frac{e^{-x/2}}{\sqrt{2\pi x}}$$