4.6 Deviation: Markov & Chebyshev Bounds

Variance


  1. What is the maximum variance of an indicator variable?

    Exercise 1

    \(p(1-p)\) maximized for \(p=0.5\).

  2. Given independent random variables \(X\) and \(Y\), what is \(Var(aX + bY + c)\)?

    Exercise 2
    • \(Var(X+a)=Var(X)\) for any constant \(a\).
    • \(Var(aX)=a^2Var(X)\) for any constant \(a\).
    • \(Var(X+Y)=Var(X)+Var(Y)\) for independent random variables \(X\) and \(Y\).

    Applying those three rules we get that \(Var(aX + bY + c)=a^2\cdot Var(X) + b^2\cdot Var(Y)\)