FRM Part I · FRM Exam Part I · Multivariate Random Variables
Random variables X and Y have joint distribution: P(X=1,Y=10)=0.10, P(X=1,Y=20)=0.30, P(X=2,Y=10)=0.30, P(X=2,Y=20)=0.30. What is E[Y | X=1]?
E[Y given X=1] is 17.5. Given X=1, the marginal probability is 0.40, so Y equals 10 with probability 0.25 and 20 with probability 0.75. The conditional expectation is 10 times 0.25 plus 20 times 0.75, which equals 17.5.
- A15.0
- B17.5Correct
- C12.5
- D17.0
Explanation
P(X=1)=0.40. Conditional probabilities: P(Y=10|X=1)=0.25 and P(Y=20|X=1)=0.75. E[Y|X=1]=10(0.25)+20(0.75)=2.5+15=17.5. The value 15 wrongly treats the outcomes as equally likely.
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