Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

514 Optimal Tests of Hypotheses

8.5.6.LetXandY have a joint bivariate normal distribution. An observation

(x, y) arises from the joint distribution with parameters equal to either

μ′ 1 =μ′ 2 =0, (σ^21 )′=(σ^22 )′=1,ρ′=^12

or

μ′′ 1 =μ′′ 2 =1, (σ^21 )′′=4, (σ 22 )′′=9,ρ′′=^12.

Show that the classification rule involves a second-degree polynomial inxandy.

8.5.7. LetW′ =(W 1 ,W 2 ) be an observation from one of two bivariate normal
distributions, I and II, each withμ 1 =μ 2 = 0 but with the respective variance-
covariance matrices

V 1 =

(

10

04

)

and V 2 =

(

30

012

)

.

How would you classifyWinto I or II?