What would be the distribution of $y$, when:
- $y = x^2$ and $x\sim\mathcal{N}(\mu, \sigma^2)$.
- $y = x^2$ and $x\sim$ Log-$\mathcal{N}$.
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1$\begingroup$ is $\mathbb{N}$ suppose to denote the standard normal distribution? If so, the first one is $\chi^{2}_{1}$. $\endgroup$– MacroCommented Jul 18, 2011 at 2:07
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2$\begingroup$ Sounds like homework..? $\endgroup$– hawkCommented Jul 18, 2011 at 12:40
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2 Answers
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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 $X\sim\mathcal{N}\left(0,1\right)$.
answered Jul 18, 2011 at 2:22
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Max answered the second question.
For the first question, $y=x^2$ is a Non-central chi-square distibution, up to a scalar ($\sigma^2$).