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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 .