APFIVE Minimizing Variance of a Linear Combination
Two independent random variables X and Y have the following properties: $$E(X) = 12$$, $$E(Y) = 8$$, $$\text{Var}(X) = 16$$, and $$\text{Var}(Y) = 25$$. Let $$Z = aX + bY$$ where $$a$$ and $$b$$ are constants. If $$E(Z) = 20$$ and $$\text{Var}(Z)$$ is minimized, what are the values of $$a$$ and $$b$$?
A
a = 375/289, b = 4000/7225
B
a = 1, b = 1
C
a = 2, b = -1/2
D
a = 1, b = 1/2
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