Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / O / 5 / ii / Solution

The variables $(m-1)S_X^2/{\sigma }^2$ and $(n-1)S_Y^2/{\sigma }^2$ are independent chi-squared variables with $m-1$ and $n-1$ degrees of freedom. Independence follows because the two whole samples are independent. Their sum is chi-squared with the sum of the degrees of freedom, soWithin each normal sample the sample mean is independent of its sample variance, and independence of the samples makes the pair of means jointly independent of the pair of variances. Hence this pooled chi-squared variable is independent of the standardized mean difference, which is the independence required for the Student pivot.