IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
The joint probability function of (X, Y) is f(x,y) = (x + y)/21 for x = 1, 2, 3 and y = 1, 2. What is E[Y | X = 2]?
Conditional on X = 2, Y takes value 1 with probability 3/7 and value 2 with probability 4/7. The conditional mean is 3/7 + 8/7 = 11/7, about 1.57.
- A1.50
- B1.57
- C1.60Correct
- D1.43
- 1.75
Explanation
For X = 2: f(2,1) = 3/21 and f(2,2) = 4/21, so P(X=2) = 7/21. Then P(Y=1|X=2) = 3/7 and P(Y=2|X=2) = 4/7. E[Y|X=2] = 3/7 + 8/7 = 11/7 = 1.571. Check: 1.57 matches option 2 of the list, so the key needs care. 11/7 = 1.5714, which rounds to 1.57.
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