42. Let X... . . , Xm and Y" . .. , y" be independently distributed as Na , ( 2 ) and N(T/, 'T 2 ) , respectively and let (t T/ ,

a, T) have the joint improper prior density 7Ta ,1J,

a, 'T) d~ d1J do ds = d~ d1J (l/a) do (1/'T) dr. Extend the result of Example 15 to inferences concerning 'T 2/ a2 • Note. The posterior distribution of 'II - in this case is the so-called Behrens-Fisher distribution. The credible regions for T/ - obtained from this distribution do not correspond to confidence intervals with fixed coverage probability, and the associated tests of H :'II = thus do not have fixed size (which instead depends on T/a). From numerical evidence [see Robinson (1976) for a summary of his and earlier results] it appears that the confidence intervals are conservative, that is, the actual coverage probability always exceeds the nominal one.