IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
X and Y are independent and each uniformly distributed on (0,1). Let S = X + Y. What is P(S > 1.5)?
The probability is 1/8. The region x + y > 1.5 in the unit square is a right triangle with legs of length 0.5, so its area is 0.5 x 0.5 x 0.5 = 0.125, and the joint density is 1 on the square.
- A1/4
- B1/8Correct
- C1/2
- D3/8
- 1/16
Explanation
The sum has a triangular density. P(S > 1.5) is the area of the corner of the unit square where x + y > 1.5: a right triangle with legs 0.5, area 0.5 x 0.5 x 0.5 = 0.125 = 1/8. Option 1/4 forgets the factor 1/2 of the triangle area.
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