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I am having trouble understanding the proof for variance portion below. Which rule did the author apply for variance? Does not look familiar to the standard variance formulas I have learnt. And what does the subscript x in this case mean, since for the said variance equation, there is a subscript of x only for one of the V[.] parts. Thanks.

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The author used the total variance formula stating that $\mathrm{var}(Z) = \mathrm{var}_X(E(Z|X))+E_X( \mathrm{var}(Z|X))$ with $Z=uX$. Then conditional variance and expectations are easy to compute.

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