Find the pdf of $Y=e^{X^2}$, where $X\sim N(0,1)$

Question

$Y=e^{X^2}$ , where $X\sim N(0,1)$

I want to find the pdf of $Y$.

Answer 1

Guide:

Find CDF of $Y$ for positive $y$ on base of: $$P(Y\leq y)=P(X^2\leq\ln y)=P(-(\ln y)^{\frac12}\leq X\leq(\ln y)^{\frac12})=2\Phi((\ln y)^\frac12)-1$$

Then find PDF by differentiating.