Math 5090
Quiz 4
Name:
Show all work clearly and in order, and circle or box your final answers.
1.) Let X1 , X2 , . . . , Xn denote an i.i.d. sequence of random variables with
density
f (x, θ1 ) =
1 −x/θ1
e
,
θ1
and let Y1 , Y2 , . . . , Ym denote an i.i.d. sequence of random variables with
density
f (x, θ2 ) =
1 −x/θ2
e
.
θ2
.
Assume that the Xi ’s are independent of the Yj ’s. Find an approximate
size α GLR test of H0 : θ1 = θ2 versus H0 : θ1 6= θ2 .
2.) Let X1 , X2 , . . . , Xn denote an i.i.d. sequence of random variables with
density
1
−(x−µ1 )2 /2σ12
f (x, µ1 , σ1 ) = p
e
,
2πσ12
and let Y1 , Y2 , . . . , Ym denote an i.i.d. sequence of random variables with
density
1
−(x−µ2 )2 /2σ22
f (x, µ2 , σ2 ) = p
e
2πσ22
.
Assume that the Xi ’s are independent of the Yj ’s, and that µ1 , µ2 , σ1 , and
σ2 are unknown. Find an approximate size α GLR test of H0 : µ1 = µ2
versus H0 : µ1 6= µ2 .