 Prediction interval in a normal linear model

For the full-rank normal linear model $Y=X\beta +\epsilon$ and an independent future response $Y^{\ast }=x^{\ast T}\beta +{\epsilon }^{\ast }$, putThen an exact $(1-\alpha )$ prediction interval isThe additional one inside the square root is the future observation's irreducible error variance; a confidence interval for its mean omits it.