IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
X and Y have joint density f(x,y)=x+y for 0<x<1, 0<y<1. What is E[Y | X=0.5]?
E[Y | X=0.5] is 0.583. The marginal density of X at 0.5 equals 1, so the conditional density of Y is 0.5+y on (0,1). Integrating y times this gives 0.25 plus one third, which is 0.583.
- A0.500
- B0.542
- C0.583Correct
- D0.611
- 0.667
Explanation
Marginal f_X(x)=x+1/2, which is 1 at x=0.5. Conditional density of Y is (0.5+y)/1. E[Y|X=0.5]=∫y(0.5+y)dy=0.25+1/3=0.5833. Check: 0.5/2=0.25 and 1/3=0.3333, sum 0.5833.
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