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

X and Y satisfy E[Y|X] = 2 + 3X. X has mean 4 and variance 5. Var(Y|X) = 10 for all X. What is Var(Y)?

Var(Y) is 55. The expected conditional variance is 10, and the variance of the conditional mean 2 + 3X is 9 times Var(X) = 45. By the law of total variance, adding them gives 55.

  1. A10
  2. B15
  3. C45
  4. D55Correct
  5. 65

Explanation

E[Var(Y|X)] = 10. Var(E[Y|X]) = 3² × 5 = 45. Total = 55. Option 45 omits the conditional variance term.

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