Question

Suppose Y1 and Y2 are independent normal with same variance. (a) Show that U1 = Y1 +Y2 and U2 = Y1 - Y2 are joint normal. (b) Show that U1 = Y1 +Y2 and U2 = Y1 - Y2 are independent.

Answer 1

Joint distribution calculation and checking their independence

4. Y , &Y are independently distinted normal variables with same rarrance, her ya N(M1002) indent Yg > indeptyly N(H2/02) (a) = Y, 7 & U2 = Y, -12 The joint distribution of Y, &Y is as follows: frixe (73) t hen ) (3-4+(83*). U = Y,+42 U2 Y1+Y 2 Y, z. Ui+U2 Y,= vitu? 272 : UI-W2 → Y22 V, -V2 Jacobian of transformation is 1J1 f dy, au dy, auz dys ayz du at uz 1 |- |-- N Join-distribution of U,,Uz is as follows: fuu,(a, ve) - fra *) 41

olaco + 1782 424 uuz - (W1+42)M, + ,2 4 ,82 - (4, -22) M2 + 4,12} S . 2 + 202 12 - Holli+M2) - U2 (Hor Mr. zh 2702 - hot () 12 Hoteca (5-my-Go-Roma)} 2 fu (ai 1 Mid210) & fur (U21 M1, M2,0) where, fu, (ki 1 M, M240) t c-Box - (45*4237 Tres

< 262932 { , - (9403) 2 = UN N( (4, 44), (+20)²), (") & fur (uz 1 M1, M2,0) a c-3600) 41 - 61-493? & Van N( (41-42), (207?). (6) By the representation (4) we can see that U, & va jointly follower roomal distribution giren by in & (6) respectively, and as fuiuz = fu, xfuz U & We are independently normal or independently distributed ,