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Q21Section A (Section A)21 of 60

Variance from a joint MGF

Let (X,Y)(X,Y) be a random vector with joint moment generating function M(t1,t2)=1(1−(t1+t2))(1−t2),−∞<t1<∞,−∞<t2<min⁡{1,1−t1}.M(t_{1},t_{2})=\frac{1}{\left(1-(t_{1}+t_{2})\right)\left(1-t_{2}\right)},\quad-\infty<t_{1}<\infty, -\infty<t_{2}<\min\{1,1-t_{1}\}. Let Z=X+YZ=X+Y. Then, Var⁡(Z)\operatorname{Var}(Z) is equal to

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