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What is the distribution of the square of a non-standard normal random variable (i.e., the mean is not equal to 0 and the variance is not equal to 1)?

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It is a scaled non-central chi-square distribution with one degree of freedom. More specifically, if $Z$ is a normal random variable with mean $\mu$ and variance $\sigma^2$, then $\frac{Z^2}{\sigma^2}$ is a non-central chi-square random variable with one degree of freedom and non-centrality parameter $\lambda=\left(\frac{\mu}{\sigma}\right)^2$.

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  • $\begingroup$ The non-centrality parameter in the non-central chi square is the square of the mean of the normal distribution in question. What is the scaling factor that we multiply the non-central chi square by to account for the variance of the normal distribution not being equal to 1? $\endgroup$ – Thomas Mar 13 '16 at 19:44
  • $\begingroup$ Just made an edit to address this. $\endgroup$ – Brent Kerby Mar 13 '16 at 20:12
  • $\begingroup$ To make it even more concrete for those of us who like concrete, to generate m random values of Z squared, in R you can use Z2 <- s^2 * rchisq(m,df=1,ncp=(mu/s)^2) $\endgroup$ – Thomas Mar 14 '16 at 9:31

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