IAI Actuarial Core Principles · Actuarial Statistics · Expectations and conditional expectations
X and Y have joint density f(x,y) = x + y on 0<x<1, 0<y<1. What is E[Y | X = 1/2]?
E[Y | X=1/2] is 7/12. The marginal density of X at 1/2 is 1, and the integral of y times (1/2 + y) over 0 to 1 is 1/4 + 1/3 = 7/12. Y is not independent of X, so 1/2 is wrong.
- A7/12Correct
- B1/2
- C5/8
- D2/3
- 3/5
Explanation
f_X(1/2) = ∫(1/2 + y)dy = 1/2 + 1/2 = 1. Numerator ∫y(1/2 + y)dy = 1/4 + 1/3 = 7/12. So E[Y|X=1/2] = 7/12. The value 1/2 would result if Y were assumed independent of X.
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