I'm trying to determine the distribution of $Z^2$ where $Z \sim N(0,1)$. I'm not really sure even how to start, is it ok to just multiply the density functions? May I need to use the MGF?
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Possible Duplicate – Mark Viola Sep 10 '15 at 17:03
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possible duplicate of Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution – Sep 10 '15 at 22:15
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You can't just multiply the density functions, but you can calculate the CDF of $Z^2$ in terms of $\Phi(x) = P(Z \le x)$. Note that $$P(Z^2 \le x) = P(|Z| \le \sqrt{x}) = P(-\sqrt{x} \le Z \le \sqrt{x}) = \Phi(\sqrt{x}) - \Phi(-\sqrt{x})$$ holds for any $x \ge 0$. From this you can calculate the density by deriving the CDF.
The distribution of $Z^2$ is called chi-squared distribution with $1$ degree of freedom. The corresponding notation is $Z^2 \sim \chi^2_1$.
