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

The joint probability function of X and Y is P(X=0,Y=0)=0.2, P(X=0,Y=1)=0.1, P(X=1,Y=0)=0.3, P(X=1,Y=1)=0.4. What is E[X | Y = 1]?

E[X | Y=1] is 0.8. The marginal probability P(Y=1) is 0.5, and the probability that X=1 within that is 0.4/0.5 = 0.8. Since X takes values 0 and 1, its conditional mean equals that probability.

  1. A0.5
  2. B0.6
  3. C0.7
  4. D0.8Correct
  5. 0.4

Explanation

P(Y=1) = 0.1 + 0.4 = 0.5. Given Y=1, P(X=1) = 0.4/0.5 = 0.8, so E[X|Y=1] = 0.8. Using 0.4 forgets to divide by P(Y=1); 0.7 is the unconditional P(X=1).

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