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Suppose there are n independent Gaussian r.v.s, X j ? N ( ? j , ?...

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Basic Math

Suppose there are n independent Gaussian r.v.s, X j ? N ( ? j ,? 2 j ) for j = 1 , 2 , . . . , n , with possibly different meansand variances

(a)  For any constants a j’s, find the MGF of a linearcombination of these n independent Gaussian r.v.s, i.e., the MGFofY=a1X1+a2X2+· · ·anXn(=Pnj=1ajXj)

Hint: Since MX(t) =E(etX), it is the case that E(eatX) =MX(at),i.e., the MGF evaluated at at.

(b) Recall that the MGF uniquely defines the distribution of arandom variable, i.e., if two random variables have the same MGF,then their distributions must be the same. Based on the functionalform of the MGF of Y in (a), specify the name of Y’s distribution,including the parameter

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