Properties of Expectation and Variance (continuous version)

Proposition

Let X be a continuous rv with PDF f(x) and mean μ and standard deviation σ. Then:

  1. (variance shortcut): Var(X)=E(X2)−μ2
  2. (Variance and Standard Deviation of X#^fc0c98): For any constant k≥1:
P(|X−μ|≥kσ)≤1k2
  1. (linearity of expectation): For any functions h1(X),h2(X) and any constants a1,a2,b:
E[a1h1(X)+a2h2(X)+b]=a1E[h1(X)]+a2E[h2(X)]+b
  1. (rescaling) For any constants a,b:
E(aX+b)=aμ+b,Var(aX+b)=a2σ2,σaX+b=|a|σ

All the proofs of these come from the discrete cases.