Proof

Zi=Xiμi/σi,i=1,2si193_e are independent N0,1,si194_e and X1+X2=μ1+μ2+σ1Z1+σ2Z2.si195_e Transform Y1Y2=σ1σ2σ2σ1Z1Z2.si196_e Then detσ1σ2σ2σ1=σ12+σ22,andZ1Z2=1σ12+σ22σ1σ2σ2σ1Y1Y2,si197_e and the joint pdf of Y1 = σ1Z1 + σ2Z2 and Y2 = σ2Z1σ1Z2 is

fY1,Y2y1,y2=12πσ12+σ2

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