I was able to differentiate the equation for the second time to obtain the expression but I don't know how to use this to show that \(VAR[Di-Vi]=0 \). Please help!

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I don't think you've read the question correctly, you should be proving that:

$$ Var[Di - \mu Vi] = E[Di] $$

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To go about that you use the fact that:

$$ Var[Di - \mu Vi] = E[(Di- \mu Vi)^2] - (E[Di- \mu Vi])^2 $$

Where the second term is naturally 0 as you've proved in statement (4.2)

The first term is what you attain from splitting up the pdf into the two values that Di takes on: 0 or 1, and then making the substitution from taking the derivative of the expression given twice.

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