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IAI Actuarial Core Principles · Actuarial Statistics · Expectations and conditional expectations

X and Y are random variables with E[Y|X] = 3 + 2X, E[X] = 5 and Var(X) = 4. What is E[Y] and Cov(X,Y)?

E[Y]=13 and Cov(X,Y)=8. By the tower property E[Y]=3+2×5=13, and Cov(X,Y)=Cov(X,3+2X)=2Var(X)=8.

  1. AE[Y]=13, Cov=8Correct
  2. BE[Y]=13, Cov=4
  3. CE[Y]=10, Cov=8
  4. DE[Y]=13, Cov=16
  5. E[Y]=8, Cov=8

Explanation

E[Y]=E[E[Y|X]]=3+2(5)=13. Cov(X,Y)=Cov(X,E[Y|X])=2Var(X)=8. Cov of 16 wrongly squares the slope-type factor.

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