# Distribution of the square of a non-standard normal random variable

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)?

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 $\lambda=\left(\frac{\mu}{\sigma}\right)^2$.