What are the distributions of these random variables?

Question

What would be the distribution of $y$, when:

1. $y = x^2$ and $x\sim\mathcal{N}(\mu, \sigma^2)$.
2. $y = x^2$ and $x\sim$ Log-$\mathcal{N}$.

Answer 1

Assuming $X\sim\mathcal{N}\left(0,1\right)$ then $Y=X^2\sim\mathcal{\chi}_{1}^{2}$.

Assuming $X\sim\text{log-}\mathcal{N}\left(0,1\right)$ then $Y=X^2\sim\text{log-}\mathcal{N}\left(0,4\right)$.

EDIT:

In general, if $X\sim\text{log-}\mathcal{N}\left(\mu,\sigma^2\right)$, then according to the Wikipedia article, $X^\alpha\sim\text{log-}\mathcal{N}\left(\alpha\mu,\alpha^2\sigma^2\right)$.

I'm unsure of the general case for $X^\alpha$ when $X\sim\mathcal{N}\left(0,1\right)$.

Answer 2

Max answered the second question.
For the first question, $y=x^2$ is a Non-central chi-square distibution, up to a scalar ($\sigma^2$).