AppliedMultiple choice

Which of these identities requires the variables to be independent?

  1. AE[X+Y]=E[X]+E[Y]\mathbb{E}[X+Y] = \mathbb{E}[X] + \mathbb{E}[Y], whatever the dependence between them
  2. BVar⁡(X+Y)=Var⁡(X)+Var⁡(Y)\operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y)
  3. CE[aX]=a E[X]\mathbb{E}[aX] = a\,\mathbb{E}[X] for any constant aa
  4. DE[X+c]=E[X]+c\mathbb{E}[X + c] = \mathbb{E}[X] + c for a constant cc

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Linearity of expectation

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Which of these identities requires the variables to be independent? · QuantMax