Noncentral t-distribution ($t_{\nu }(\delta )$)

If $Z\sim N(0,1)$ and $U\sim {\chi }_{\nu }^2$ are independent, then $(Z+\delta )/\sqrt{U/\nu }$ has the noncentral t-distribution with degrees of freedom $\nu$ and noncentrality $\delta$. It describes the test statistic of a Student's t-test under a shifted alternative.