In this paper, we show that the classic paired t-test is a central Behrens-Fisher problem with a non-zero population correlation coefficient, derive its corresponding probability density function by solving its associated non-central Behrens-Fisher problem with a non-zero population correlation coefficient   and  and give two examples in data analysis. A more general non-central Behrens-Fisher problem with a non-zero population correlation coefficient  and  is solved in Appendix A. We also contrast the k-th moments of the standard normal distribution, the classic central t distribution, and our generalized central Behrens-Fisher distribution, in order to explain their differing effectiveness in hypothesis testing. Finally, we solve a more generalized non-central Behrens-Fisher problem with a variance-covariance    matrix for a linear combination  of m means  from a sample of size n from a multivariate normal distribution.

paired t-test, generalized Behrens-Fisher problem, generalized Behrens-Fisher distribution, central and non-central t distributions.