The distribution of the variables that originates from monitoring the variance when
the mean encountered a sustained shift is considered — specifically for the case when
measurements from each sample are independent and identically distributed normal
random variables. It is shown that the solution to this problem involves a sequence
of dependent random variables that are constructed from independent noncentral chi-
squared random variables. This sequence of dependent random variables are the key
to understanding the performance of the process used to monitor the variance and
are the focus of this article. For simplicity, the marginal (i.e. the univariate and
bivariate) distributions and the joint (i.e. the trivariate) distribution of only the first
three random variables following a change in the variance is considered. A multivariate
generalization is proposed which can be used to calculate the entire run-length (i.e.
the waiting time until the first signal) distribution.