5.19. (Sec. 5.3) Let x and S be based on N observations from N(fl., ::£), and let x be an additional observation from N(fl., ::£). Show that x - x is distributed according to N[0,(1+1IN)::£].

Verify that [N I(N + l)](x - X)'S-I(X - x) has the T2-distribution with N - 1 degrees of freedom. Show how this statistic can be used to give a prediction region for x based on x and S (i.e., a region such that one has a given confidence that the next observation will fall into it).