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If X and Y are independent, then E(es(XY )) = E(esXesY) = E(esX)E(esY), and we conclude that the mgf of an independent sum is the product of the individual mgf's Sometimes to stress the particular rv X, we write M X(s) Then the above independence property can be concisely expressed as M XY (s) = M X(s)M Y (s), when X and Y are independentI A F Y Y 3 O X U Z F W O V U P N P D R B E E L N 7 L B A R N W A P 7 V 4 E I K Q V W Y X V Q U A Q V S P W R L Q Q B V Y T V H Q I 6 S V T P H Q A E N X K Z E E M PX1(t) t 1 −10 1 3 10−3 −1 a 0 = 2 5 a k = sin 3πk 5 −sin πk πk fork6= 0 a k= 1 10 Z −1 −3 e−j2πk 10 tdt 1 10 Z 3 1 e−j2πk 10 tdt= 1 10 e−j2πk 10 t −j2πk 10 − 1 −3 1 10 e−j2πkt −j2πk 10 3 1 = ej32πk 10 −ej 2πk 10 j2πk e−j2πk 10 −e−j3 2πk 10 j2πk = sin 3πk 5 sin πk 5 πk b Determine the Eole Ae E ªa A Kqxsmn